Function of several complex variables

3827: 8038: 3094: 3822:{\displaystyle {\begin{aligned}f(z_{1},\ldots ,z_{n})&={\frac {1}{2\pi i}}\int _{\partial D_{1}}{\frac {f(\zeta _{1},z_{2},\ldots ,z_{n})}{\zeta _{1}-z_{1}}}\,d\zeta _{1}\\&={\frac {1}{(2\pi i)^{2}}}\int _{\partial D_{2}}\,d\zeta _{2}\int _{\partial D_{1}}{\frac {f(\zeta _{1},\zeta _{2},z_{3},\ldots ,z_{n})}{(\zeta _{1}-z_{1})(\zeta _{2}-z_{2})}}\,d\zeta _{1}\\&={\frac {1}{(2\pi i)^{n}}}\int _{\partial D_{n}}\,d\zeta _{n}\cdots \int _{\partial D_{2}}\,d\zeta _{2}\int _{\partial D_{1}}{\frac {f(\zeta _{1},\zeta _{2},\ldots ,\zeta _{n})}{(\zeta _{1}-z_{1})(\zeta _{2}-z_{2})\cdots (\zeta _{n}-z_{n})}}\,d\zeta _{1}\end{aligned}}} 7435: 24500:.), but it is not easy to verify which compact complex analytic spaces are algebraizable. In fact, Hopf found a class of compact complex manifolds without nonconstant meromorphic functions. However, there is a Siegel result that gives the necessary conditions for compact complex manifolds to be algebraic. The generalization of the Riemann-Roch theorem to several complex variables was first extended to compact analytic surfaces by Kodaira, Kodaira also extended the theorem to three-dimensional, and n-dimensional Kähler varieties. Serre formulated the Riemann–Roch theorem as a problem of dimension of 8033:{\displaystyle {\begin{aligned}\omega (z)&=\sum _{k=0}^{\infty }{\frac {1}{k!}}{\frac {1}{(2\pi i)^{n}}}\int _{|\zeta _{\nu }|=R_{\nu }}\cdots \int \omega (\zeta )\times \left_{z=0}df_{\zeta }\cdot z^{k}\\&+\sum _{k=1}^{\infty }{\frac {1}{k!}}{\frac {1}{2\pi i}}\int _{|\zeta _{\nu }|=r_{\nu }}\cdots \int \omega (\zeta )\times \left(0,\cdots ,{\sqrt {\frac {k!}{\alpha _{1}!\cdots \alpha _{n}!}}}\cdot \zeta _{n}^{\alpha _{1}-1}\cdots \zeta _{n}^{\alpha _{n}-1},\cdots 0\right)df_{\zeta }\cdot {\frac {1}{z^{k}}}\ (\alpha _{1}+\cdots +\alpha _{n}=k)\end{aligned}}} 5874: 13039: 4626: 5371: 4176: 8742: 21936:, every holomorphic function on it is constant by Liouville's theorem, and so it cannot have any embedding into complex n-space. That is, for several complex variables, arbitrary complex manifolds do not always have holomorphic functions that are not constants. So, consider the conditions under which a complex manifold has a holomorphic function that is not a constant. Now if we had a holomorphic embedding of 5869:{\displaystyle {\begin{aligned}&f(z)=\sum _{k_{1},\dots ,k_{n}=0}^{\infty }c_{k_{1},\dots ,k_{n}}(z_{1}-a_{1})^{k_{1}}\cdots (z_{n}-a_{n})^{k_{n}}\ ,\\&c_{k_{1}\cdots k_{n}}={\frac {1}{(2\pi i)^{n}}}\int _{\partial D_{1}}\cdots \int _{\partial D_{n}}{\frac {f(\zeta _{1},\dots ,\zeta _{n})}{(\zeta _{1}-a_{1})^{k_{1}+1}\cdots (\zeta _{n}-a_{n})^{k_{n}+1}}}\,d\zeta _{1}\cdots d\zeta _{n}\end{aligned}}} 9974:

doesn't always hold for any domain. Therefore, in order to study of the domain of convergence of the power series, it was necessary to make additional restriction on the domain, this was the Reinhardt domain. Early knowledge into the properties of field of study of several complex variables, such as Logarithmically-convex, Hartogs's extension theorem, etc., were given in the Reinhardt domain.

4621:{\displaystyle {\frac {\partial ^{k_{1}+\cdots +k_{n}}f(\zeta _{1},\zeta _{2},\ldots ,\zeta _{n})}{\partial {z_{1}}^{k_{1}}\cdots \partial {z_{n}}^{k_{n}}}}={\frac {k_{1}\cdots k_{n}!}{(2\pi i)^{n}}}\int _{\partial D_{1}}\cdots \int _{\partial D_{n}}{\frac {f(\zeta _{1},\dots ,\zeta _{n})}{(\zeta _{1}-z_{1})^{k_{1}+1}\cdots (\zeta _{n}-z_{n})^{k_{n}+1}}}\,d\zeta _{1}\cdots d\zeta _{n}.} 5174: 6605: 4139: 2253: 8454: 14915:, so it was named by Levi as a pseudoconvex domain (Hartogs's pseudoconvexity). Pseudoconvex domain (boundary of pseudoconvexity) are important, as they allow for classification of domains of holomorphy. A domain of holomorphy is a global property, by contrast, pseudoconvexity is that local analytic or local geometric property of the boundary of a domain. 3038: 20174:, which satisfies these conditions, is one way to define a Stein manifold. The study of the cousin's problem made us realize that in the study of several complex variables, it is possible to study of global properties from the patching of local data, that is it has developed the theory of sheaf cohomology. (e.g.Cartan seminar.) 1973: 9030:. This result can be proven from the fact that holomorphics functions have power series extensions, and it can also be deduced from the one variable case. Contrary to the one variable case, it is possible that two different holomorphic functions coincide on a set which has an accumulation point, for instance the maps 20158:

complex coordinate space, also solving the second Cousin problem with additional topological assumptions. The Cousin problem is a problem related to the analytical properties of complex manifolds, but the only obstructions to solving problems of a complex analytic property are pure topological; Serre called this the

4857: 22835: 4903: 6346: 2083: 13030:. Cartan and more development Serre. In sheaf cohomology, the domain of holomorphy has come to be interpreted as the theory of Stein manifolds. The notion of the domain of holomorphy is also considered in other complex manifolds, furthermore also the complex analytic space which is its generalization. 24811:

But there is a point where it converges outside the circle of convergence. For example if one of the variables is 0, then some terms, represented by the product of this variable, will be 0 regardless of the values taken by the other variables. Therefore, even if you take a variable that diverges when

12920:

When moving from the theory of one complex variable to the theory of several complex variables, depending on the range of the domain, it may not be possible to define a holomorphic function such that the boundary of the domain becomes a natural boundary. Considering the domain where the boundaries of

24608:

embeds as an algebraic variety. This result gives an example of a complex manifold with enough meromorphic functions. Broadly, the GAGA principle says that the geometry of projective complex analytic spaces (or manifolds) is equivalent to the geometry of projective complex varieties. The combination

9973:

In polydisks, the Cauchy's integral formula holds and the power series expansion of holomorphic functions is defined, but polydisks and open unit balls are not biholomorphic mapping because the Riemann mapping theorem does not hold, and also, polydisks was possible to separation of variables, but it

3875: 24929:

In algebraic geometry, there is a problem whether it is possible to remove the singular point of the complex analytic space by performing an operation called modification on the complex analytic space (when n = 2, the result by Hirzebruch, when n = 3 the result by Zariski for algebraic varietie.),

20157:

are not isolated points; these problems are called the Cousin problems and are formulated in terms of sheaf cohomology. They were first introduced in special cases by Pierre Cousin in 1895. It was Oka who showed the conditions for solving first Cousin problem for the domain of holomorphy on the

11992: 8737:{\displaystyle \omega (\zeta ,z)={\frac {(n-1)!}{(2\pi i)^{n}}}{\frac {1}{|z-\zeta |^{2n}}}\sum _{1\leq j\leq n}({\overline {\zeta }}_{j}-{\overline {z}}_{j})\,d{\overline {\zeta }}_{1}\land d\zeta _{1}\land \cdots \land d\zeta _{j}\land \cdots \land d{\overline {\zeta }}_{n}\land d\zeta _{n}} 583:

is known as a similar result for compact complex manifolds, and the Grauert–Riemenschneider conjecture is a special case of the conjecture of Narasimhan. In fact it was the need to put (in particular) the work of Oka on a clearer basis that led quickly to the consistent use of sheaves for the

12095:

It is also called Osgood–Brown theorem is that for holomorphic functions of several complex variables, the singularity is a accumulation point, not an isolated point. This means that the various properties that hold for holomorphic functions of one-variable complex variables do not hold for

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of compact complex manifolds has developed as Kodaira–Spencer theory. However, despite being a compact complex manifold, there are counterexample of that cannot be embedded in projective space and are not algebraic. Analogy of the Levi problems on the complex projective space

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Note that the Riemann extension theorem and its references explained in the linked article includes a generalized version of the Riemann extension theorem by Grothendieck that was proved using the GAGA principle, also every one-dimensional compact complex manifold is a Hodge

6209: 1829: 14153: 11547: 6868: 21677: 20637: 15782: 19125: 18839:) (Especially, coherent analytic sheaf) in sheaf cohomology. This name comes from H. Cartan. Also, Serre (1955) introduced the notion of the coherent sheaf into algebraic geometry, that is, the notion of the coherent algebraic sheaf. The notion of coherent ( 10742: 226:, i.e. the problem of creating a global meromorphic function from zeros and poles, is called the Cousin problem. Also, the interesting phenomena that occur in several complex variables are fundamentally important to the study of compact complex manifolds and 22376: 16313:

is positive definite at every point. The strongly pseudoconvex domain is the pseudoconvex domain. Strongly pseudoconvex and strictly pseudoconvex (i.e. 1-convex and 1-complete) are often used interchangeably, see Lempert for the technical difference.

21020: 15583: 10554: 19804: 2353: 5169:{\displaystyle \left|{\frac {\partial ^{k_{1}+\cdots +k_{n}}f(\zeta _{1},\zeta _{2},\ldots ,\zeta _{n})}{{\partial z_{1}}^{k_{1}}\cdots \partial {z_{n}}^{k_{n}}}}\right|\leq {\frac {Mk_{1}\cdots k_{n}!}{{r_{1}}^{k_{1}}\cdots {r_{n}}^{k_{n}}}}} 16680: 4688: 22674: 21439: 6600:{\displaystyle {\frac {\partial ^{k_{1}+\cdots +k_{n}}f}{\partial {z_{1}}^{k_{1}}\cdots \partial {z_{n}}^{k_{n}}}}=\sum _{v=1}^{\infty }{\frac {\partial ^{k_{1}+\cdots +k_{n}}f_{v}}{\partial {z_{1}}^{k_{1}}\cdots \partial {z_{n}}^{k_{n}}}}} 24599:

shows that the complex analytic subspace (subvariety) of a closed complex projective space to be an algebraic that is, so it is the common zero of some homogeneous polynomials, such a relationship is one example of what is called Serre's

12635: 20504: 24396:

Stein manifolds are in some sense dual to the elliptic manifolds in complex analysis which admit "many" holomorphic functions from the complex numbers into themselves. It is known that a Stein manifold is elliptic if and only if it is

22099:

and satisfies the separation condition described later, the condition for becoming a Stein manifold is to satisfy the holomorphic convexity. Therefore, the Stein manifold is the properties of the domain of definition of the (maximal)

2839: 4134:{\displaystyle f(z_{1},\dots ,z_{n})={\frac {1}{(2\pi i)^{n}}}\int _{\partial D_{1}}\cdots \int _{\partial D_{n}}{\frac {f(\zeta _{1},\dots ,\zeta _{n})}{(\zeta _{1}-z_{1})\cdots (\zeta _{n}-z_{n})}}\,d\zeta _{1}\cdots d\zeta _{n}} 9926:

as the natural boundary. There is called the Hartogs's phenomenon. Therefore, researching when domain boundaries become natural boundaries has become one of the main research themes of several complex variables. In addition, when

19594: 11287: 24443:). In fact, compact Riemann surface had a non-constant single-valued meromorphic function, and also a compact Riemann surface had enough meromorphic functions. A compact one-dimensional complex manifold was a Riemann sphere 8969: 2248:{\displaystyle \forall i\in \{1,\dots ,n\},\quad {\frac {\partial u}{\partial x_{i}}}={\frac {\partial v}{\partial y_{i}}}\quad {\text{ and }}\quad {\frac {\partial u}{\partial y_{i}}}=-{\frac {\partial v}{\partial x_{i}}}} 11777: 15429: 11819: 11170: 2735: 6214:

We have already explained that holomorphic functions on polydisc are analytic. Also, from the theorem derived by Weierstrass, we can see that the analytic function on polydisc (convergent power series) is holomorphic.

14522: 22940: 19440: 2661: 24508:. Cartan and Serre proved the following property: the cohomology group is finite-dimensional for a coherent sheaf on a compact complex manifold M. Riemann–Roch on a Riemann surface for a vector bundle was proved by 16288: 16201: 16130: 8748: 7047: 6873: 1658: 10794: 7265: 16687: 12495: 5963: 15098: 19922: 12385: 10198: 3099: 3033:{\displaystyle {\overline {\Delta }}(z,r)=\left\{\zeta =(\zeta _{1},\zeta _{2},\dots ,\zeta _{n})\in \mathbb {C} ^{n};\left|\zeta _{\nu }-z_{\nu }\right|\leq r_{\nu }{\text{ for all }}\nu =1,\dots ,n\right\}} 24524:.). Next, the generalization of the result that "the compact Riemann surfaces are projective" to the high-dimension. In particular, consider the conditions that when embedding of compact complex submanifold 24490: 1141: 21031: 18790: 14978: 12759: 5198: 12840: 6004: 23720: 17553: 17435: 7440: 7221:

In this way it is possible to have a similar, combination of radius of convergence for a one complex variable. This combination is generally not unique and there are an infinite number of combinations.

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was able to create a global meromorphic function from a given zeroes or zero-locus (Cousin II problem). However, these theorems do not hold in several complex variables because the singularities of

17655: 10190: 8103: 24019: 16403: 14001: 24172: 18646: 13646: 13210: 222:) that are not the domain of holomorphy of any function, and so is not always the domain of holomorphy, so the domain of holomorphy is one of the themes in this field. Patching the local data of 23409: 18233: 5376: 1528: 220: 21529: 11360: 6681: 2018: 1440: 446: 22662: 20880: 16055: 14622: 10433: 17291: 17024: 15249: 9644: 20064: 17351: 6676: 2264: 1732: 24072: 20519: 21237: 15662: 13700: 13597: 19014: 15961: 15460: 15371: 15020: 14663: 10598: 8043:

The integral in the second term, of the right-hand side is performed so as to see the zero on the left in every plane, also this integrated series is uniformly convergent in the annulus

8189: 8146: 1968:{\displaystyle {\frac {\partial u}{\partial x}}(p)={\frac {\partial v}{\partial y}}(p)\quad {\text{ and }}\quad {\frac {\partial u}{\partial y}}(p)=-{\frac {\partial v}{\partial x}}(p)} 22230: 21290: 18158: 13878: 13315: 13106: 10010: 9016: 1349: 21810: 18038: 17078: 16959: 4676: 22990: 21734: 17708: 15139: 24604:. The complex analytic sub-space(variety) of the complex projective space has both algebraic and analytic properties. Then combined with Kodaira's result, a compact Kähler manifold 22465: 20029: 19218: 18904: 17829:

was a domain of holomorphy (holomorphically convex), it was proved that it was necessary and sufficient that each boundary point of the domain of holomorphy is an Oka pseudoconvex.

16513: 14574: 11809: 9901: 9862: 9557: 9518: 2395: 21521: 21363: 14395: 13476: 12909: 2765: 166: 9145: 1216: 24305: 23846: 14850: 6263: 25231: 24353: 23515: 23202: 21473: 20131: 19986: 19859: 19680: 19625: 19304: 18959: 18551: 18381: 16880: 16434: 16356: 15912: 15816: 15634: 13543: 11593: 3082: 2539: 2495: 359: 172:

of some function, in other words every domain has a function for which it is the domain of holomorphy. For several complex variables, this is not the case; there exist domains (

27262:

Ohsawa, Takeo (February 2021). "NISHIno's Rigidity, Locally pseudoconvex maps, and holomorphic motions (Topology of pseudoconvex domains and analysis of reproducing kernels)".

24648: 24558: 22551: 22174: 21901: 20908: 18588: 15507: 14193: 13955: 13801: 10442: 9590: 9348: 8384: 1760: 994: 260: 27650: 25149: 25084: 24896: 23790: 23758: 23548: 23091: 22093: 22033: 21996: 21967: 21930: 21319: 20830: 20801: 20093: 19955: 18348: 18075: 17827: 16472: 15187: 14897: 14758: 14266: 13373: 13250: 13010: 12981: 12948: 12264: 12210: 12183: 12129: 12052: 11661: 11355: 11323: 9479: 9274: 9174: 9083: 8349: 8241: 1297: 1268: 959: 878: 845: 816: 755: 561: 289: 59: 24243: 23231: 17864: 16546: 13341: 11046: 22512: 19828: 19696: 19269: 19242: 18928: 18311: 18285: 18259: 17145: 16928: 14359: 2593: 2460: 18007: 17790: 17743: 9744: 27226: 25171: 24441: 21853: 19489: 17942: 14701: 9306: 8199:

The Cauchy integral formula holds only for polydiscs, and in the domain of several complex variables, polydiscs are only one of many possible domains, so we introduce the

4852:{\displaystyle \left\{\zeta =(\zeta _{1},\zeta _{2},\dots ,\zeta _{n})\in \mathbb {C} ^{n};|\zeta _{\nu }-z_{\nu }|\leq r_{\nu },{\text{ for all }}\nu =1,\dots ,n\right\}} 1371: 926: 787: 521: 27552:

Hans Grauert & Reinhold Remmert (1956), "Konvexität in der komplexen Analysis. Nicht-holomorph-konvexe Holomorphiegebiete und Anwendungen auf die Abbildungstheorie",

22830:{\displaystyle H^{1}(X,{\mathcal {O}}_{X})\longrightarrow H^{1}(X,{\mathcal {O}}_{X}^{*})\longrightarrow H^{2}(X,\mathbb {Z} )\longrightarrow H^{2}(X,{\mathcal {O}}_{X})} 20710: 20278: 18825: 18463: 18077:

the locally pseudoconvex domain is itself a pseudoconvex domain and it is a domain of holomorphy. For example, Diederich–Fornæss found local pseudoconvex bounded domains

12237: 12156: 4895: 23590: 23469: 23132: 23038: 22219: 14791: 13985: 13280: 13133: 10786: 10590: 9924: 9791: 8264: 7257: 3853: 16574: 24800: 23338: 21371: 19645: 17387: 17047: 15995: 2566: 23286: 22422: 20769: 20335: 18503: 18115: 18095: 17902: 17455: 14729: 14424: 13834: 13060: 12860: 11001:

When a some complete Reinhardt domain to be the domain of convergence of a power series, an additional condition is required, which is called logarithmically-convex.

10036: 9951: 9694: 9207: 8446: 2433: 23921: 21760: 19515: 17968: 16311: 15498: 12502: 9418: 8408: 8316: 8284: 1397: 24108: 23967: 23617: 20900: 20413: 20366: 20238: 20211: 17209: 17189: 17165: 17114: 13761: 13734: 12709: 12682: 9368: 6329: 6298: 24517: 23260: 17088:

When the Levi (–Krzoska) form is positive-definite, it is called strongly Levi (–Krzoska) pseudoconvex or often called simply strongly (or strictly) pseudoconvex.

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If such a relations holds in the domain of holomorphy of several complex variables, it looks like a more manageable condition than a holomorphically convex. The

9440:. However, even in the case of several complex variables, there are some results similar to the results of the theory of uniformization in one complex variable. 24911:

This is a hullomorphically convex hull condition expressed by a plurisubharmonic function. For this reason, it is also called p-pseudoconvex or simply p-convex.

22141: 20730: 20386: 20298: 19460: 19367: 19347: 19324: 19168: 19148: 19003: 18983: 18483: 16566: 15654: 15603: 15277: 14323: 14220: 13918: 13898: 9819: 9664: 2770: 2078: 2058: 2038: 1824: 1800: 1780: 24177:

Related to the previous item, another equivalent and more topological definition in complex dimension 2 is the following: a Stein surface is a complex surface

11634:

When examining the domain of convergence on the Reinhardt domain, Hartogs found the Hartogs's phenomenon in which holomorphic functions in some domain on the

28464:

Fornaess, J.E.; Forstneric, F; Wold, E.F (2020). "The Legacy of Weierstrass, Runge, Oka–Weil, and Mergelyan". In Breaz, Daniel; Rassias, Michael Th. (eds.).

24930:

but, Grauert and Remmert has reported an example of a domain that is neither pseudoconvex nor holomorphic convex, even though it is a domain of holomorphy:

24417:) holds for compact Riemann surfaces (Therefore the theory of compact Riemann surface can be regarded as the theory of (smooth (non-singular) projective) 22574: 19520: 11181: 11987:{\displaystyle H_{\varepsilon }=\{z=(z_{1},z_{2})\in \Delta ^{2};|z_{1}|<\varepsilon \ \cup \ 1-\varepsilon <|z_{2}|\}\ (0<\varepsilon <1)} 28564:

Stein, Karl (1951), "Analytische Funktionen mehrerer komplexer Veränderlichen zu vorgegebenen Periodizitätsmoduln und das zweite Cousinsche Problem",

24834:

This theorem holds even if the condition is not restricted to the bounded. i.e. The theorem holds even if this condition is replaced with an open set.

8894: 26142: 13212:, the boundaries may not be natural boundaries. Hartogs' extension theorem gives an example of a domain where boundaries are not natural boundaries. 11670: 15379: 11054: 2666: 29267: 26019: 18313:, i.e. constructing a global holomorphic function which admits no extension from non-extendable functions defined only locally. This is called the 8874:{\displaystyle \displaystyle f(z)=\int _{\partial D}f(\zeta )\omega (\zeta ,z)-\int _{D}{\overline {\partial }}f(\zeta )\land \omega (\zeta ,z).} 7211:{\displaystyle \left\{z=(z_{1},z_{2},\dots ,z_{n})\in \mathbb {C} ^{n};|z_{\nu }-a_{\nu }|>r_{\nu },{\text{ for all }}\nu =1,\dots ,n\right\}} 7037:{\displaystyle \left\{z=(z_{1},z_{2},\dots ,z_{n})\in \mathbb {C} ^{n};|z_{\nu }-a_{\nu }|<r_{\nu },{\text{ for all }}\nu =1,\dots ,n\right\}} 1675: 27474:

Friedrich Hirzebruch (1953), "Über vierdimensionaleRIEMANNsche Flächen mehrdeutiger analytischer Funktionen von zwei komplexen Veränderlichen",

14437: 31281: 22847: 19372: 2598: 1010:

In coordinate-free language, any vector space over complex numbers may be thought of as a real vector space of twice as many dimensions, where

26368:

Hans J. Bremermann (1954), "Über die Äquivalenz der pseudokonvexen Gebiete und der Holomorphiegebiete im Raum vonn komplexen Veränderlichen",

16230: 16143: 16072: 10959:{\displaystyle \left\{z=(z_{1},\dots ,z_{n});\left|z_{\nu }-a_{\nu }\right|\leq \left|z_{\nu }^{0}-a_{\nu }\right|,\ \nu =1,\dots ,n\right\}.} 7422:{\displaystyle \left\{z=(z_{1},z_{2},\dots ,z_{n})\in \mathbb {C} ^{n};r_{\nu }<|z|<R_{\nu },{\text{ for all }}\nu +1,\dots ,n\right\}} 1552: 580: 23166:

A Complex analytic space which admits a continuous strictly plurisubharmonic exhaustion function (i.e.strongly pseudoconvex) is Stein space.

16837:{\displaystyle H(\rho )=\sum _{i,j=1}^{n}{\frac {\partial ^{2}\rho (p)}{\partial z_{i}\,\partial {\bar {z_{j}}}}}w_{i}{\bar {w_{j}}}\geq 0.} 24825:, which is a generalization of the convergence domain, a Reinhardt domain is a domain of holomorphy if and only if logarithmically convex. 18407:

Oka introduced the notion which he termed "idéal de domaines indéterminés" or "ideal of indeterminate domains". Specifically, it is a set

10363:{\displaystyle \left\{z=(z_{1},\dots ,z_{n});\left|z_{\nu }-a_{\nu }\right|=\left|z_{\nu }^{0}-a_{\nu }\right|,\ \nu =1,\dots ,n\right\}.} 12392: 5889: 27874:"Éléments de géométrie algébrique: I. Le langage des schémas (ch.0 § 5. FAISCEAUX QUASI-COHÉRENTS ET FAISCEAUX COHÉRENTS (0.5.1–0.5.3))" 15039: 698:

can be derived from analytic functions. In a sense this doesn't contradict Siegel; the modern theory has its own, different directions.

408: 19877: 12287: 30805: 29162:

Eliashberg, Yakov; Gromov, Mikhael (1992). "Embeddings of Stein Manifolds of Dimension n into the Affine Space of Dimension 3n/2 +1".

24446: 21172:{\displaystyle H^{0}(M,\mathbf {K} ^{*}){\xrightarrow {\phi }}H^{0}(M,\mathbf {K} ^{*}/\mathbf {O} ^{*})\to H^{1}(M,\mathbf {O} ^{*})} 5358:{\displaystyle \{z=(z_{1},z_{2},\dots ,z_{n})\in \mathbb {C} ^{n};|z_{\nu }-a_{\nu }|<r_{\nu },{\text{ for all }}\nu =1,\dots ,n\}} 1087: 470:(since we are in four real dimensions), while iterating contour (line) integrals over two separate complex variables should come to a 28095: 23419:

This means that Behnke–Stein theorem, which holds for Stein manifolds, has not found a conditions to be established in Stein space.

18384: 9220:, and implicit function theorems also hold. For a generalized version of the implicit function theorem to complex variables, see the 6204:{\displaystyle f(z)=\sum _{k_{1},\dots ,k_{n}=0}^{\infty }c_{k_{1},\dots ,k_{n}}(z_{1}-a_{1})^{k_{1}}\cdots (z_{n}-a_{n})^{k_{n}}\ ,} 5180: 26171:"Zu den Abbildungen durch analytische Funktionen mehrerer komplexer Veränderlichen die Invarianz des Mittelpunktes von Kreiskörpern" 21859:

into the complex plane. In other words, there is a holomorphic mapping into the complex plane whose derivative never vanishes.) The

24601: 14929: 12714: 12764: 24513: 23859:

These facts imply that a Stein manifold is a closed complex submanifold of complex space, whose complex structure is that of the

23622: 17460: 17396: 27824: 15287: 10041: 30823: 26332:

Oka, Kiyoshi (1953), "Sur les fonctions analytiques de plusieurs variables. IX. Domaines finis sans point critique intérieur",

18350:

by Kiyoshi Oka, but for ramified Riemann domains, pseudoconvexity does not characterize holomorphically convexity, and then by

14148:{\displaystyle {\hat {K}}_{G}:=\left\{z\in G;|f(z)|\leq \sup _{w\in K}|f(w)|{\text{ for all }}f\in {\mathcal {O}}(G).\right\}.} 30895: 15825: 31271: 31117: 31017: 30877: 30648: 30595: 30569: 30492: 30472: 30377: 30350: 30274: 30216: 30197: 29898: 29865: 29793: 29519: 29477: 28495: 28359: 28233: 28129: 28078: 28051: 27938: 27791: 27748: 26846: 26757: 25902: 25826: 25791: 25524: 25283: 25016: 24409:

Meromorphic function in one-variable complex function were studied in a compact (closed) Riemann surface, because since the

20154: 17562: 10114: 26804: 23980: 16365: 11542:{\textstyle \sum _{k_{1},\dots ,k_{n}=0}^{\infty }c_{k_{1},\dots ,k_{n}}(z_{1}-a_{1})^{k_{1}}\cdots (z_{n}-a_{n})^{k_{n}}\ } 6863:{\textstyle \sum _{k_{1},\dots ,k_{n}=0}^{\infty }c_{k_{1},\dots ,k_{n}}(z_{1}-a_{1})^{k_{1}}\cdots (z_{n}-a_{n})^{k_{n}}\ } 2362: 30733: 26696:

Chorlay, Renaud (January 2010). "From Problems to Structures: the Cousin Problems and the Emergence of the Sheaf Concept".

24121: 21825:

Since a non-compact (open) Riemann surface always has a non-constant single-valued holomorphic function, and satisfies the

21672:{\displaystyle H^{1}(M,\mathbf {O} )\to H^{1}(M,\mathbf {O} ^{*})\to 2\pi iH^{2}(M,\mathbb {Z} )\to H^{2}(M,\mathbf {O} ).} 13605: 30841: 30787: 13162: 30967: 30103:"Takeuchi's equality for the levi form of the Fubini–Study distance to complex submanifolds in complex projective spaces" 25641: 23363: 18726: 18206: 1447: 175: 30364:. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics. Vol. 56. 26833:. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics. Vol. 56. 24370:

Numerous further characterizations of such manifolds exist, in particular capturing the property of their having "many"

16140:-dimensional manifold. Then is said to be weeak pseudoconvex there exists a smooth plurisubharmonic exhaustion function 12281:

to one of the following domains provided that the orbit of the origin by the automorphism group has positive dimension:

1980: 1402: 364: 31350: 31264:

Henri Cartan et André Weil. Mathématiciens du XXesiècle. Journées mathématiques X-UPS, Palaiseau, France, May 3–4, 2012

30697: 27181: 24509: 22603: 21855:. (In fact, Gunning and Narasimhan have shown (1967) that every non-compact Riemann surface actually has a holomorphic 20839: 16004: 14579: 10383: 78: 30751: 21762:

so that a necessary and sufficient condition in that case for the second Cousin problem to be always solvable is that

20632:{\displaystyle H^{0}(M,\mathbf {K} ){\xrightarrow {\phi }}H^{0}(M,\mathbf {K} /\mathbf {O} )\to H^{1}(M,\mathbf {O} )} 18654: 18593: 17235: 16972: 15196: 9599: 31128:"Scientific report on the second summer institute, several complex variables. Part I. Report on the analysis seminar" 30931: 30769: 30677: 30621: 30542: 30513: 30446: 30396: 30316: 30295: 30238: 29955:

Kodaira, K. (1954). "On Kahler Varieties of Restricted Type (An Intrinsic Characterization of Algebraic Varieties)".

29442: 29415: 29070: 27973: 27019: 26992: 26253:"Zur Theorie der Singularitäten der Funktionen mehrerer komplexen Veränderlichen Regularitäts-und Konvergenzbereiche" 25711: 20150: 20036: 18399:

into several complex variables allowed the reformulation of and solution to several important problems in the field.

17296: 15777:{\displaystyle H_{u}=(\lambda _{ij}),\lambda _{ij}={\frac {\partial ^{2}u}{\partial z_{i}\,\partial {\bar {z}}_{j}}}} 6618: 30949: 30859: 25965:

Sitzungsberichte der Königlich Bayerischen Akademie der Wissenschaften zu München, Mathematisch-Physikalische Klasse

25086:

is a projective complex varieties) does not become a Stein manifold, even if it satisfies the holomorphic convexity.

19120:{\displaystyle {\mathcal {O}}_{X}^{\oplus I}|_{U}\to {\mathcal {O}}_{X}^{\oplus J}|_{U}\to {\mathcal {F}}|_{U}\to 0} 13810:

is always solvable in a domain of holomorphy, also Cartan showed that the converse of this result was incorrect for

10737:{\displaystyle \left\{z=(z_{1},\dots ,z_{n});z=a+\left(z^{0}-a\right)e^{i\theta },\ 0\leq \theta <2\pi \right\}.} 2020:

is holomorphic if and only if it is holomorphic in each variable separately, and hence if and only if the real part

1701: 227: 28251:"Sur les fonctions analytiques de plusieurs variables. I. Domaines convexes par rapport aux fonctions rationnelles" 27594:

Tsurumi, Kazuyuki; Jimbo, Toshiya (1969). "Some properties of holomorphic convexity in general function algebras".

25554: 24568:

in 1953) gives the condition, when the sheaf cohomology group vanishing, and the condition is to satisfy a kind of

24024: 20280:

is holomorphic (wherever the difference is defined). The first Cousin problem then asks for a meromorphic function

8046: 30913: 30715: 29594:

Kodaira, Kunihiko (1952). "The Theorem of Riemann-Roch for Adjoint Systems on 3-Dimensional Algebraic Varieties".

29305: 27451: 26140:

Cartan, Henri (1931). "Les fonctions de deux variables complexes et le problème de la représentation analytique".

22371:{\displaystyle {\bar {K}}=\left\{z\in X;|f(z)|\leq \sup _{w\in K}|f(w)|,\ \forall f\in {\mathcal {O}}(X)\right\},} 21185: 31355: 27684:

Noguchi, Junjiro (2019). "A brief chronicle of the Levi (Hartog's inverse) problem, coherence and open problem".

24689: 20510: 15929: 15437: 15344: 14988: 14627: 13651: 13548: 12097: 9221: 1234: 332: 17083: 31182:"Scientific report on the second summer institute, several complex variables. Part III. Algebraic sheaf theory" 24493: 21244: 20712:

is a non-vanishing holomorphic function (where said difference is defined). It asks for a meromorphic function

18128: 13848: 13285: 13076: 12954:. Cartan and Thullen. Levi's problem shows that the pseudoconvex domain was a domain of holomorphy. (First for 9980: 9667: 8986: 8200: 1319: 137: 26566:

Oka, Kiyoshi (1951), "Sur les Fonctions Analytiques de Plusieurs Variables, VIII--Lemme Fondamental (Suite)",

26288: 21765: 18012: 17052: 16933: 13487: 12950:

call the domain of holomorphy), the first result of the domain of holomorphy was the holomorphic convexity of

4650: 30977: 30959: 30941: 30923: 30905: 30887: 30869: 30851: 30833: 30815: 30797: 30779: 30761: 30743: 30725: 30707: 29315: 27461: 27191: 25651: 22945: 21689: 17660: 15103: 1027: 604: 600: 30334:

Complex Analysis 2: Riemann Surfaces, Several Complex Variables, Abelian Functions, Higher Modular Functions

26862:

Behnke, H.; Stein, K. (1939). "Konvergente Folgen von Regularitätsbereichen und die Meromorphiekonvexität".

25005:

In the case of Stein space with isolated singularities, it has already been positively solved by Narasimhan.

22431: 19991: 19180: 18866: 16477: 14535: 11782: 9867: 9828: 9523: 9484: 5365:, from the Cauchy's integral formula, we can see that it can be uniquely expanded to the next power series. 27334: 26458:"Sur les formes objectives et les contenus subjectifs dans les sciences math'ematiques; Propos post'erieur" 24375: 22841: 21826: 21482: 21324: 20655: 20134: 14368: 13453: 13017: 12865: 12096:

holomorphic functions of several complex variables. The nature of these singularities is also derived from

10986: 8151: 8108: 3085: 2743: 1803: 143: 26803:

Cartan, H.; Bruhat, F.; Cerf, Jean.; Dolbeault, P.; Frenkel, Jean.; Hervé, Michel; Malatian.; Serre, J-P.

26457: 21015:{\displaystyle H^{0}(M,\mathbf {K} ^{*}){\xrightarrow {\phi }}H^{0}(M,\mathbf {K} ^{*}/\mathbf {O} ^{*}).} 20149:

was able to create a global meromorphic function from a given and principal parts (Cousin I problem), and

15578:{\displaystyle \Delta =4\left({\frac {\partial ^{2}u}{\partial z\,\partial {\overline {z}}}}\right)\geq 0} 10549:{\displaystyle \left\{z^{0}-a_{\nu }\right\}\to \left\{e^{i\theta _{\nu }}(z_{\nu }^{0}-a_{\nu })\right\}} 9088: 1221:

Likewise, if one expresses any finite-dimensional complex linear operator as a real matrix (which will be

1155: 523:

we can find a function that will nowhere continue analytically over the boundary, that cannot be said for

31155:"Scientific report on the second summer institute, several complex variables. Part II. Complex manifolds" 30972: 30954: 30936: 30918: 30900: 30882: 30864: 30846: 30828: 30810: 30792: 30774: 30756: 30738: 30720: 30702: 29310: 28387:

Heinrich Behnke & Karl Stein (1948), "Entwicklung analytischer Funktionen auf Riemannschen Flächen",

27456: 27186: 26293: 25808: 25646: 24259: 23813: 23012: 20146: 18402: 15975: 14796: 6222: 1230: 592: 31208: 30025:

Calabi, Eugenio; Eckmann, Beno (1953). "A Class of Compact, Complex Manifolds Which are not Algebraic".

25207: 24310: 23474: 23181: 21447: 20670:

The second Cousin problem starts with a similar set-up to the first, specifying instead that each ratio

20105: 19960: 19833: 19654: 19599: 19278: 18933: 18512: 18357: 17832: 16854: 16408: 16330: 15886: 15790: 15608: 13502: 11552: 3043: 2500: 2469: 31244: 28651: 27622: 27405: 26969: 24621: 24531: 22665: 22523: 22146: 22058:

is similar to a Stein manifold but is allowed to have singularities. Stein spaces are the analogues of

21874: 19799:{\displaystyle 0\to {\mathcal {F}}_{1}|_{U}\to {\mathcal {F}}_{2}|_{U}\to {\mathcal {F}}_{3}|_{U}\to 0} 14165: 13927: 13766: 9562: 9311: 8354: 2348:{\displaystyle \forall i\in \{1,\dots ,n\},\quad {\frac {\partial f}{\partial {\overline {z_{i}}}}}=0,} 1737: 1226: 1049: 967: 881: 695: 386: 340: 233: 27626: 26545:

Oka, Kiyoshi (1951), "Sur les Fonctions Analytiques de Plusieurs Variables, VIII--Lemme Fondamental",

25125: 25060: 24872: 23766: 23734: 23524: 23067: 22069: 22009: 21972: 21943: 21906: 21295: 20806: 20777: 20642:

is exact, and so the first Cousin problem is always solvable provided that the first cohomology group

20069: 19931: 19870:(Oka–Cartan) coherent theorem says that each sheaf that meets the following conditions is a coherent. 18324: 18051: 17803: 16439: 15148: 14855: 14734: 14229: 13346: 13226: 12986: 12957: 12924: 12242: 12188: 12161: 12107: 12028: 11637: 11331: 11299: 9455: 9250: 9150: 9033: 8325: 8217: 1273: 1244: 935: 854: 821: 792: 731: 537: 497:. Here a major difference is evident from the one-variable theory; while for every open connected set 493:, quickly changed the picture of the theory. A number of issues were clarified, in particular that of 265: 35: 24573: 24565: 24501: 24497: 24196: 23946: 23207: 22593: 22395: 21860: 18840: 17840: 16522: 13921: 13320: 11015: 2356: 706: 680: 26521: 25960: 22470: 19809: 19250: 19223: 18909: 18290: 18264: 18238: 17126: 16900: 14328: 9770:

it is unique. When n > 2, the following phenomenon occurs depending on the shape of the boundary

2571: 2438: 24733: 24588: 20389: 17973: 17760: 17713: 15030: 13703: 11621:. But, there is an example of a complete Reinhardt domain D which is not logarithmically convex. 9699: 9217: 1399:

if it is complex-differentiable at this point, in the sense that there exists a complex linear map

889: 726: 320: 30391:, Grundlehren der Mathematischen Wissenschaften, vol. 236, Berlin-New York: Springer-Verlag, 29885:. Lecture Notes in Mathematics. Vol. 225. Springer Science+Business Media. pp. xii+700. 27209: 25154: 24424: 21836: 21365:

with its additive structure by taking a logarithm. That is, there is an exact sequence of sheaves

19468: 18795:

The origin of indeterminate domains comes from the fact that domains change depending on the pair

18556: 17915: 16675:{\displaystyle \nabla \rho (p)w=\sum _{i=1}^{n}{\frac {\partial \rho (p)}{\partial z_{j}}}w_{j}=0} 14684: 9279: 1354: 909: 770: 504: 27518:

Oscar Zariski (1944), "Reduction of the Singularities of Algebraic Three Dimensional Varieties",

25273: 24969:

There are some counterexamples in the domain of holomorphicity regarding second Cousin problem.

24414: 21434:{\displaystyle 0\to 2\pi i\mathbb {Z} \to \mathbf {O} \xrightarrow {\exp } \mathbf {O} ^{*}\to 0} 20832:

the sheaf of meromorphic functions that are not identically zero. These are both then sheaves of

20673: 20243: 20182:

Without the language of sheaves, the problem can be formulated as follows. On a complex manifold

18798: 18436: 12215: 12134: 12070: 4862: 1011: 25664:

Ozaki, Shigeo; Onô, Isao (February 1, 1953). "Analytic Functions of Several Complex Variables".

23553: 23441: 23111: 23017: 22198: 14763: 13964: 13259: 13159:, which is holomorphic, cannot be extended any more. For several complex variables, i.e. domain 13115: 10758: 10562: 10559:

The Reinhardt domains which are defined by the following condition; Together with all points of

9906: 9773: 8246: 7233: 3835: 31248: 27730: 26522:"Sur les fonctions analytiques de plusieurs variables. VII. Sur quelques notions arithmétiques" 26496:"Sur les fonctions analytiques de plusieurs variables. VII. Sur quelques notions arithmétiques" 25961:"Einige Folgerungen aus der Cauchyschen Integralformel bei Funktionen mehrerer Veränderlichen." 23159: 12630:{\displaystyle \{(z,w)\in \mathbb {C} ^{2};~|z|^{2}+|w|^{\frac {2}{p}}<1\}\,(p>0,\neq 1)} 12277:'s classical result says that a 2-dimensional bounded Reinhard domain containing the origin is 11009: 29497: 29432: 29397: 29052: 28221: 26774: 24778: 23311: 20499:{\displaystyle H^{0}(M,\mathbf {K} ){\xrightarrow {\phi }}H^{0}(M,\mathbf {K} /\mathbf {O} ).} 19630: 17356: 17032: 15980: 15259:, the notion can be defined on an arbitrary complex manifold or even a Complex analytic space 9232:

From the establishment of the inverse function theorem, the following mapping can be defined.

2544: 29459: 29054:

L2 Approaches in Several Complex Variables: Towards the Oka–Cartan Theory with Precise Bounds

28068: 28041: 27957:

L2 Approaches in Several Complex Variables: Towards the Oka–Cartan Theory with Precise Bounds

27955: 27920: 27773: 26826: 26739: 25884: 25773: 23265: 22401: 22101: 20739: 20307: 18488: 18100: 18080: 17881: 17440: 17084:

Strongly (or Strictly) Levi (–Krzoska) pseudoconvex (a.k.a. Strongly (Strictly) pseudoconvex)

14714: 14403: 13813: 13045: 12845: 10015: 9930: 9673: 9425: 9179: 8983:, as in one variable : two holomorphic functions defined on the same connected open set 8413: 7260: 2418: 2258: 1035: 997: 962: 494: 402: 328: 316: 27009: 26982: 26779:

Centre Belge Rech. Math., Colloque Fonctions Plusieurs Variables, Bruxelles du 11 Au 14 Mars

25549: 23900: 21739: 19494: 18354:

using methods from functional analysis and partial differential equations (a consequence of

17947: 16293: 15468: 13836:. this is also true, with additional topological assumptions, for the second Cousin problem. 10985:, also when the complete Reinhardt domain is the boundary line, there is a way to prove the 9397: 8393: 8301: 8269: 1376: 377:

in the 1930s, a general theory began to emerge; others working in the area at the time were

31027: 30406: 30332: 29941: 29724: 29642: 29529: 28636: 28585: 28369: 27897: 27859: 26646: 25585: 25445: 25387: 24694: 24674: 24410: 24371: 24093: 23952: 23595: 22035:

are called Stein manifolds. Also Stein manifolds satisfy the second axiom of countability.

20885: 20344: 20216: 20189: 18040:-plurisubharmonic exhaustion function (weakly 1-complete), in this situation, we call that 17194: 17174: 17150: 17099: 14908: 13739: 13712: 12687: 12660: 12005: 10990: 9353: 6307: 6276: 1238: 710: 685: 664: 398: 223: 169: 97: 31223: 31043: 29762: 29545: 29214: 28901: 28768: 28416: 27581: 27437: 27131: 26786: 26654: 26314: 26155: 25976: 25593: 25453: 23236: 22517: 15465:

In one-complex variable, necessary and sufficient condition that the real-valued function

5968: 2834:{\displaystyle {\overline {\Delta }}\subset {D_{1}\times D_{2}\times \cdots \times D_{n}}} 1229:

of the corresponding complex determinant. It is a non-negative number, which implies that

327:

is a candidate. The theory, however, for many years didn't become a full-fledged field in

8: 29746: 29198: 28991:

Coltoiu, Mihnea (2009). "The Levi problem on Stein spaces with singularities. A survey".

28607: 28345: 27990: 25096: 24853: 24609:

of analytic and algebraic methods for complex projective varieties lead to areas such as

24492:. However, the abstract notion of a compact Riemann surface is always algebraizable (The 18835: 18410: 18396: 18180: 15972: 14282: 13483: 13427: 13394: 11598: 1687: 1536:

is said to be holomorphic if it is holomorphic at all points of its domain of definition

668: 336: 117: 29811:"Zur algebraischen Theorie der algebraischen Funktionen. (Aus einem Brief an H. Hasse.)" 29646: 29199:"Sur les espaces analytiques holomorphiquement séparables et holomorphiquement convexes" 12086:

regardless of how it is chosen can be each extended to a unique holomorphic function on

7429:

and continuous on their circumference, then there exists the following expansion ;

31301: 31256: 31096: 31068: 31031: 30654: 30157: 30083: 30042: 30007: 29972: 29830: 29728: 29673: 29660: 29630: 29611: 29576: 29533: 29378: 29286: 29179: 29144: 29109: 29033: 28992: 28966: 28931: 28889: 28872:

Hans Grauert (1958), "On Levi's Problem and the Imbedding of Real-Analytic Manifolds",

28847: 28812: 28794: 28671: 28622: 28589: 28546: 28528: 28501: 28473: 28446: 28404: 28351: 28189:"Sur les fonctions analytiques de plusieurs variables. III–Deuxième problème de Cousin" 28023: 28005: 27901: 27847: 27797: 27754: 27711: 27693: 27603: 27569: 27535: 27501: 27425: 27353: 27310: 26879: 26721: 26713: 26438: 26420: 26387: 26233: 26192: 26089: 26069: 26038: 25993: 25935: 25865: 25750: 25673: 25489: 25433: 25325: 25254: 24614: 24356: 24114:

to the idea of a corresponding class of compact complex manifolds with boundary called

22126: 20715: 20371: 20283: 19925: 19445: 19352: 19332: 19309: 19153: 19133: 18988: 18968: 18468: 16551: 16058: 15639: 15588: 15262: 14308: 14205: 13903: 13883: 13479: 12100:. A generalization of this theorem using the same method as Hartogs was proved in 2007. 9804: 9649: 9433: 2063: 2043: 2023: 1809: 1785: 1765: 585: 296: 29559:

Kodaira, Kunihiko (1951). "The Theorem of Riemann-Roch on Compact Analytic Surfaces".

29281: 29262: 29104: 29087: 28740: 28723: 28316: 26033: 26014: 24577: 21833:-dimensional complex manifold possessing a holomorphic mapping into the complex plane 18403:

Idéal de domaines indéterminés (The predecessor of the notion of the coherent (sheaf))

12185:

is the Reinhardt domain containing the center z = 0, and the domain of convergence of

885: 569:). The natural domains of definition of functions, continued to the limit, are called 31321: 31267: 31113: 31035: 31013: 30673: 30658: 30644: 30617: 30591: 30565: 30538: 30509: 30505:

Holomorphic Functions of Several Variables: An Introduction to the Fundamental Theory

30488: 30468: 30442: 30418: 30392: 30373: 30346: 30312: 30291: 30270: 30253: 30234: 30212: 30193: 30161: 30136:"Domaines pseudoconvexes infinis et la métrique riemannienne dans un espace projecti" 30087: 29894: 29861: 29834: 29789: 29732: 29696: 29678: 29537: 29515: 29473: 29438: 29411: 29127:

Raghavan, Narasimhan (1960). "Imbedding of Holomorphically Complete Complex Spaces".

29066: 28970: 28935: 28851: 28816: 28675: 28593: 28550: 28505: 28491: 28450: 28408: 28355: 28229: 28125: 28074: 28047: 28027: 27969: 27934: 27905: 27820: 27801: 27787: 27758: 27744: 27715: 27573: 27505: 27496: 27429: 27357: 27314: 27015: 26988: 26883: 26842: 26753: 26442: 26391: 26351: 26302: 26237: 26196: 25956: 25939: 25898: 25889:. Graduate Texts in Mathematics. Vol. 108. p. 10.1007/978-1-4757-1918-5_2. 25869: 25822: 25787: 25754: 25707: 25573: 25545: 25520: 25493: 25279: 25258: 24737: 24684: 24561: 24087: 23802:

The embedding theorem for Stein manifolds states the following: Every Stein manifold

19686: 19327: 18318: 17833:

Locally pseudoconvex (a.k.a. locally Stein, Cartan pseudoconvex, local Levi property)

12009: 11357:

is the interior of the set of points of absolute convergence of some power series in

9213: 8387: 3864: 3860: 2413: 1679: 929: 758: 626: 390: 355: 31198: 31181: 31171: 31154: 31144: 31127: 30458: 30288:

Théorie élémentaire des fonctions analytiques d'une ou plusieurs variables complexes

29246: 29229: 26725: 25992:

Simonič, Aleksander (2016). "Elementary approach to the Hartogs extension theorem".

25378: 25361: 25329: 20407:. The first Cousin problem can always be solved if the following map is surjective: 18351: 13038: 12915: 9429: 31219: 31193: 31166: 31139: 31088: 31060: 31039: 31005: 30993: 30636: 30609: 30583: 30557: 30530: 30434: 30365: 30338: 30328: 30185: 30147: 30114: 30073: 30034: 29999: 29964: 29927: 29886: 29853: 29822: 29781: 29758: 29712: 29668: 29650: 29603: 29568: 29541: 29507: 29493: 29465: 29403: 29370: 29341: 29276: 29241: 29210: 29171: 29136: 29099: 29058: 29025: 28958: 28923: 28897: 28881: 28839: 28804: 28764: 28735: 28702: 28663: 28573: 28538: 28483: 28438: 28429:

Gunning, R. C.; Narasimhan, Raghavan (1967). "Immersion of open Riemann surfaces".

28412: 28396: 28295: 28262: 28200: 28162: 28117: 28015: 27961: 27926: 27885: 27839: 27779: 27736: 27703: 27663: 27577: 27561: 27527: 27491: 27483: 27433: 27417: 27384: 27343: 27302: 27271: 27239: 27160: 27127: 27117: 27081: 27048: 26943: 26910: 26871: 26834: 26782: 26745: 26705: 26650: 26614: 26575: 26554: 26507: 26430: 26377: 26341: 26310: 26289:"Sur les fonctions analytiques de plusieurs variables. VI. Domaines pseudoconvexes" 26264: 26223: 26182: 26151: 26120: 26079: 26028: 25972: 25927: 25890: 25855: 25814: 25779: 25740: 25699: 25620: 25589: 25563: 25512: 25481: 25449: 25423: 25373: 25315: 25244: 25100: 24664: 24659: 24379: 24246: 22120: 22096: 21903:, whereas it is "rare" for a complex manifold to have a holomorphic embedding into 19651:

Morphisms between (quasi-)coherent sheaves are the same as morphisms of sheaves of

18962: 17121: 15657: 13023: 12646: 10982: 10974: 8980: 691: 676: 652: 612: 608: 596: 591:

From this point onwards there was a foundational theory, which could be applied to

576: 534:

of that kind are rather special in nature (especially in complex coordinate spaces

490: 475: 394: 85: 31079:

Seebach, J. Arthur; Seebach, Linda A.; Steen, Lynn A. (1970). "What is a Sheaf?".

28116:. Grundlehren der mathematischen Wissenschaften. Vol. 265. pp. 152–166. 26093: 24759: 23931:

which form a local coordinate system when restricted to some open neighborhood of

21815: 19589:{\displaystyle \varphi :{\mathcal {O}}_{X}^{\oplus n}|_{U}\to {\mathcal {F}}|_{U}} 14624:. Also, at this time, D is a domain of holomorphy. Therefore, every convex domain 11282:{\displaystyle \lambda ;z\rightarrow \lambda (z)=(\ln |z_{1}|,\dots ,\ln |z_{n}|)} 31023: 31001: 30667: 30524: 30503: 30482: 30462: 30402: 30306: 29937: 29720: 29525: 29503: 28632: 28581: 28469: 28365: 28284:"Sur les fonctions analytiques de plusieurs variables. II–Domaines d'holomorphie" 27893: 27873: 27855: 26808: 26669: 26642: 25703: 25581: 25441: 25383: 24717: 24418: 23879: 23064:

This was proved by Bremermann by embedding it in a sufficiently high dimensional

22996: 22570: 22054:

complex dimensions. They were introduced by and named after Karl Stein (1951). A

15336: 15280: 14912: 13807: 1057: 1015: 672: 471: 378: 348: 308: 307:

Many examples of such functions were familiar in nineteenth-century mathematics;

69:. The name of the field dealing with the properties of these functions is called 29469: 29014:"Some open problems in higher dimensional complex analysis and complex dynamics" 28121: 27930: 27775:

Analytic Function Theory of Several Variables Elements of Oka's Coherence (p.33)

25894: 25516: 22589: 13478:) was the domain of holomorphy; we can define a holomorphic function with zeros 8964:{\displaystyle \displaystyle f(z)=\int _{\partial D}f(\zeta )\omega (\zeta ,z).} 31309: 30580:

Holomorphic Functions and Integral Representations in Several Complex Variables

29635:

Proceedings of the National Academy of Sciences of the United States of America

29346: 29329: 28542: 27732:

Analytic Function Theory of Several Variables Elements of Oka's Coherence (p.x)

27707: 27389: 27372: 26125: 26108: 25886:

Holomorphic Functions and Integral Representations in Several Complex Variables

25625: 25608: 25428: 25411: 25151:, but not the only method like the Riemann sphere that was compactification of 24949: 24899: 24749:

The field of complex numbers is a 2-dimensional vector space over real numbers.

24398: 24390: 23883: 22560: 22059: 19006: 18829: 11772:{\displaystyle \Delta ^{2}=\{z\in \mathbb {C} ^{2};|z_{1}|<1,|z_{2}|<1\}} 1061: 1039: 896: 848: 714: 571: 312: 292: 74: 66: 31009: 30640: 30587: 30369: 30342: 30290:(in French) (6é. ed., nouv. tir ed.). Paris : Hermann. p. 231. 30189: 29857: 29826: 29511: 29062: 28808: 28757:

Seminars on Analytic Functions. Institute for Advanced Study (Princeton, N.J.)

28519:

Patyi, Imre (2011). "On complex Banach manifolds similar to Stein manifolds".

28487: 28300: 28283: 28267: 28250: 28205: 28188: 27965: 27783: 27740: 27108:

Sin Hitomatsu (1958), "On some conjectures concerning pseudo-convex domains",

26838: 26749: 26709: 26434: 25931: 25860: 25843: 25783: 25769: 25320: 25303: 24596: 15424:{\displaystyle f\circ \varphi \colon \Delta \to \mathbb {R} \cup \{-\infty \}} 11165:{\displaystyle D^{*}=\{z=(z_{1},\dots ,z_{n})\in D;z_{1},\dots ,z_{n}\neq 0\}} 5186: 2730:{\displaystyle {\overline {D_{1}\times D_{2}\times \cdots \times D_{n}}}\in D} 295:, these are much similar to study of algebraic varieties that is study of the 31344: 30257: 30078: 30061: 29029: 28019: 26915: 26898: 26475: 26355: 26306: 25577: 25507:

Freitag, Eberhard (2011). "Analytic Functions of Several Complex Variables".

24569: 24505: 24174:. Some authors call such manifolds therefore strictly pseudoconvex manifolds. 23887: 23860: 23849: 23427:

Grauert introduced the concept of K-complete in the proof of Levi's problem.

22063: 20833: 15868: 12278: 12274: 8287: 1079: 1053: 702: 656: 616: 456: 382: 31333: 31313: 30422: 30152: 30135: 30119: 30102: 29751:

Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences de Paris

29203:

Comptes Rendus Hebdomadaires des Séances de l'Académie des Sciences de Paris

28112:

Grauert, Hans; Remmert, Reinhold (1984). "Normalization of Complex Spaces".

27165: 27148: 27122: 27086: 27069: 26580: 26559: 26346: 23878:

In one complex dimension the Stein condition can be simplified: a connected

17796:

is called Oka pseudoconvex. Oka's proof of Levi's problem was that when the

16132:. Often, the definition of pseudoconvex is used here and is written as; Let 8194: 2408:

meets the conditions of being continuous and separately homorphic on domain

323:. Naturally also same function of one variable that depends on some complex 30308:

Elementary Theory of Analytic Functions of One or Several Complex Variables

30226: 29682: 29631:"On the Theorem of Riemann-Roch for Adjoint Systems on Kahlerian Varieties" 28341: 27153:

Memoirs of the Faculty of Science, Kyushu University. Series A, Mathematics

27074:

Memoirs of the Faculty of Science, Kyushu University. Series A, Mathematics

27070:"Stein Neighborhood Bases for Product Sets of Polydiscs and Open Intervals" 25729:"Les fonctions analytiques de deux variables et la représentation conforme" 24679: 24610: 23974: 22585: 22047: 18861: 18314: 1222: 660: 640: 486: 482: 344: 106: 31325: 31284:. Spring 2005. Massachusetts Institute of Technology: MIT OpenCourseWare, 30613: 29655: 29361:

Neeman, Amnon (1988). "Steins, Affines and Hilbert's Fourteenth Problem".

24404: 14517:{\displaystyle {\text{dist}}(K,D^{c})={\text{dist}}({\hat {K}}_{D},D^{c})} 8290:

of differential forms. Then the Bochner–Martinelli formula states that if

31305: 30633:

Analytic Function Theory of Several Variables Elements of Oka's Coherence

30233:. Graduate Text in Mathematics. Vol. 81. New-York: Springer Verlag. 30062:"On the complement of effective divisors with semipositive normal bundle" 28830:

Grauert, Hans (1963). "Bemerkenswerte pseudokonvexe Mannigfaltigkeiten".

27623:"A counterexample for the Levi problem for branched Riemann domains over 27037:"La métrique de Kobayashi et la représentation des domaines sur la boule" 24669: 22935:{\displaystyle H^{1}(X,{\mathcal {O}}_{X})=H^{2}(X,{\mathcal {O}}_{X})=0} 22117: 22043: 22006:

is just a point. Complex manifolds that can be holomorphic embedded into

19435:{\displaystyle {\mathcal {O}}_{X}^{\oplus n}|_{U}\to {\mathcal {F}}|_{U}} 16894: 14199:. The polynomially convex hull contains the holomorphically convex hull. 13013: 12025:

is a bounded (surrounded by a rectifiable closed Jordan curve) domain on

3859:

is continuous, so the order of products and sums can be exchanged so the

2656:{\displaystyle {\overline {D_{1}\times D_{2}\times \cdots \times D_{n}}}} 1147: 374: 26: 30438: 30431:

Introduction to Complex Analysis Part II. Functions of Several Variables

29464:. Encyclopaedia of Mathematical Sciences. Vol. 35. pp. 1–125. 29037: 27607: 26899:"Relations between domains of holomorphy and multiple Cousin's problems" 26717: 25677: 25472:

Chong, C.T.; Leong, Y.K. (1986). "An interview with Jean-Pierre Serre".

25437: 16283:{\displaystyle \psi \in {\text{Psh}}(X)\cap {\mathcal {C}}^{\infty }(X)} 16196:{\displaystyle \psi \in {\text{Psh}}(X)\cap {\mathcal {C}}^{\infty }(X)} 16125:{\displaystyle \psi \in {\text{Psh}}(X)\cap {\mathcal {C}}^{\infty }(X)} 12711:

are mutually biholomorphic if and only if there exists a transformation

12104:

From Hartogs's extension theorem the domain of convergence extends from

5883:

that satisfies the following conditions is called an analytic function.

1662:

are holomorphic as functions of one complex variable : we say that

31297: 31100: 31072: 30561: 30417:(in Russian). Nauka, Glav. red. fiziko-matematicheskoĭ lit-ry, Moskva. 30046: 30011: 29976: 29932: 29915: 29890: 29785: 29747:"Un théorème de finitude concernant les variétés analytiques compactes" 29716: 29615: 29580: 29382: 29290: 29183: 29148: 29113: 28962: 28949:

Narasimhan, Raghavan (1962). "The Levi problem for complex spaces II".

28927: 28893: 28843: 28707: 28690: 28667: 28577: 28442: 28400: 28167: 28146: 27925:. Encyclopaedia of Mathematical Sciences. Vol. 74. pp. 7–96. 27889: 27851: 27667: 27565: 27539: 27487: 27421: 27406:"Modifikationen komplexer Mannigfaltigkeiten und Riernannscher Gebiete" 27348: 27306: 27275: 27053: 26875: 26637:

Cartan, Henri (1953). "Variétés analytiques complexes et cohomologie".

26619: 26602: 26512: 26495: 26382: 26269: 26252: 26228: 26211: 26187: 26170: 26084: 26057: 26042: 25883:

Range, R. Michael (1986). "Domains of Holomorphy and Pseudoconvexity".

25745: 25485: 25249: 25203: 24987:

From this condition, we can see that the Stein manifold is not compact.

24857: 23852: 22516:

The open neighborhood of every point on the manifold has a holomorphic

21025:

The long exact sheaf cohomology sequence associated to the quotient is

17797: 11629: 11293: 10377:

is called a Reinhardt domain if it satisfies the following conditions:

2463: 1653:{\displaystyle z\mapsto f(z_{1},\dots ,z_{i-1},z,z_{i+1},\dots ,z_{n})} 370: 121: 31329: 29881:

Berthelot, Pierre (1971). Alexandre Grothendieck; Luc Illusie (eds.).

29700: 29407: 26744:. Grundlehren der mathematischen Wissenschaften. Vol. 136. 1990. 31266:. Palaiseau: Les Éditions de l'École Polytechnique. pp. 99–168. 30250:Введение в теорию аналитических функции многих комплексных переменных 29776:

Brînzănescu, Vasile (1996). "Vector bundles over complex manifolds".

29664: 27330:"Hartogs type extension theorems on some domains in Kähler manifolds" 27244: 27205: 26074: 21868: 13648:

is an increasing sequence of domains of holomorphy, then their union

11549:, and conversely; The domain of convergence of every power series in 9965:. This contributed to advancement of the notion of sheaf cohomology. 9437: 8210:

is a continuously differentiable function on the closure of a domain

324: 31092: 31064: 30534: 30038: 30003: 29968: 29607: 29572: 29374: 29175: 29140: 28914:

Narasimhan, Raghavan (1961). "The Levi problem for complex spaces".

28885: 28755:

Bremermann, Hans J. (1957). "On Oka's theorem for Stein manifolds".

27843: 27531: 26948: 26931: 25918:

Krantz, Steven G. (2008). "The Hartogs extension phenomenon redux".

25728: 22557:

Note that condition (3) can be derived from conditions (1) and (2).

21407: 21182:

so the second Cousin problem is solvable in all cases provided that

21074: 20951: 20555: 20449: 20098:

the normalization of the structure sheaf of a complex analytic space

31289: 30666:

Vladimirov, Vasiliy Sergeyevich; Technica, Scripta (January 2007).

29330:"A Stein domain with smooth boundary which has a product structure" 28799: 28478: 28010: 27698: 27551: 27149:"Some Results on the Equivalence of Complex-Analytic Fibre Bundles" 26603:"Idéaux et modules de fonctions analytiques de variables complexes" 25998: 25568: 24740:

learned at school. (In other words, in the sense of GAGA on Serre.)

24714: 24521: 23795:

It can be shown quite easily that every closed complex submanifold

21932:. For example, for an arbitrary compact connected complex manifold 17215:

that satisfies these conditions is called Levi total pseudoconvex.

16069:. i.e. there exists a smooth plurisubharmonic exhaustion function 12921:

the domain are natural boundaries (In the complex coordinate space

12490:{\displaystyle \{(z,w)\in \mathbb {C} ^{2};~|z|^{2}+|w|^{2}<1\}} 5958:{\displaystyle a=(a_{1},\dots ,a_{n})\in D\subset \mathbb {C} ^{n}} 2738: 467: 466:

an integral surrounding a point should be over a three-dimensional

125: 31317: 31257:"Henri Cartan et les fonctions holomorphes de plusieurs variables" 28997: 28627: 28533: 26425: 23938:

The first Cousin problem can always be solved on a Stein manifold.

21444:

where the leftmost sheaf is the locally constant sheaf with fiber

15093:{\displaystyle \{a+bz;z\in \mathbb {C} \}\subset \mathbb {C} ^{n}} 31000:. Graduate Texts in Mathematics. Vol. 52. Berlin, New York: 29990:

Chow, Wei-Liang (1949). "On Compact Complex Analytic Varieties".

29502:. Graduate Texts in Mathematics. Vol. 52. Berlin, New York: 29013: 24960:

Oka's proof uses Oka pseudoconvex instead of Cartan pseudoconvex.

24516:

generalized the theorem to compact complex manifolds in 1994 and

20162:. They are now posed, and solved, for arbitrary complex manifold 19917:{\displaystyle {\mathcal {O}}:={\mathcal {O}}_{\mathbb {C} _{n}}} 13880:

be a domain, or alternatively for a more general definition, let

12916:

Natural domain of the holomorphic function (domain of holomorphy)

12380:{\displaystyle \{(z,w)\in \mathbb {C} ^{2};~|z|<1,~|w|<1\}} 9386:

are biholomorphically equivalent or that they are biholomorphic.

4162:

Because the order of products and sums is interchangeable, from (

2399: 655:

side of the theory was subordinated to sheaves. The interest for

31051:

Krantz, Steven G. (1987), "What is Several Complex Variables?",

30433:. Translations of Mathematical Monographs. Vol. 110. 1992. 29334:

Publications of the Research Institute for Mathematical Sciences

28691:"Variétés analytiques réelles et variétés analytiques complexes" 28386: 28317:"Applications de la théorie générale à divers problèmes globaux" 27403: 27377:

Publications of the Research Institute for Mathematical Sciences

26820: 26818: 25818: 25613:

Publications of the Research Institute for Mathematical Sciences

24583:, with a Hodge metric, there is a complex-analytic embedding of 24520:

generalized it to a relative version (relative statements about

24485:{\displaystyle {\widehat {\mathbb {C} }}\cong \mathbb {CP} ^{1}} 21686:

is a Stein manifold, the middle arrow is an isomorphism because

18120: 14195:

instead to be the set of complex-valued polynomial functions on

12158:. Looking at this from the perspective of the Reinhardt domain, 6615:

It is possible to define a combination of positive real numbers

1225:

of the aforementioned form), then its determinant equals to the

1136:{\displaystyle {\begin{pmatrix}u&-v\\v&u\end{pmatrix}},} 29780:. Lecture Notes in Mathematics. Vol. 1624. pp. 1–27. 28652:"Charakterisierung der holomorph vollständigen komplexen Räume" 27686:

Notices of the International Congress of Chinese Mathematicians

26212:"Holomorphic equivalence problem for bounded Reinhardt domains" 24720: 21820: 21816:

Manifolds and analytic varieties with several complex variables

20803:

be the sheaf of holomorphic functions that vanish nowhere, and

5994:

is expressed as a power series expansion that is convergent on

29810: 27329: 27291:"A smooth pseudoconvex domain without pseudoconvex exhaustion" 27290: 27206:"On the local pseudoconvexity of certain analytic families of 27036: 8979:

Holomorphic functions of several complex variables satisfy an

262:) and has a different flavour to complex analytic geometry in 31262:. In Harinck, Pascale; Plagne, Alain; Sabbah, Claude (eds.). 31238: 30415:Введение в комплексный анализ / Vvedenie v kompleksnyĭ analiz 30211:. Princeton mathematical series. Princeton University Press. 29088:"Oka's Heftungslemma and the Levi Problem for Complex Spaces" 29079: 28228:(in French). Springer Berlin Heidelberg. p. XXIII, 598. 27922:

Several Complex Variables VII §6. Calculs of Coherent sheaves

26815: 25844:"Meromorphic or Holomorphic Completion of a Reinhardt Domain" 24843:

Oka says that the contents of these two papers are different.

24762:

for the Cauchy's integral formula on the more general domain.

23886:

it is not compact. This can be proved using a version of the

23725: 14973:{\displaystyle f\colon D\to {\mathbb {R} }\cup \{-\infty \},} 13151:

is the supremum of the domain where the holomorphic function

12754:{\displaystyle \varphi :\mathbb {C} ^{n}\to \mathbb {C} ^{n}} 10751:

is called a complete Reinhardt domain with centre at a point

30669:

Methods of the Theory of Functions of Many Complex Variables

29458:

Danilov, V. I. (1996). "Cohomology of Algebraic Varieties".

28785:

Sibony, Nessim (2018). "Levi problem in complex manifolds".

28466:

Advancements in Complex Analysis – Holomorphic Approximation

26639:

Colloque sur les fonctions de plusieurs variables, Bruxelles

26058:"A Morse-theoretical proof of the Hartogs extension theorem" 24996:

Levi problem is not true for domains in arbitrary manifolds.

22600:

is trivial. In particular, every line bundle is trivial, so

22561:

Every non-compact (open) Riemann surface is a Stein manifold

20159: 18048:. As an old name, it is also called Cartan pseudoconvex. In 16882:

boundary, the following approximation result can be useful.

12835:{\displaystyle z_{i}\mapsto r_{i}z_{\sigma (i)}(r_{i}>0)} 12212:

has been extended to the smallest complete Reinhardt domain

9389: 331:, since its characteristic phenomena weren't uncovered. The 28222:"Quelques problèmes globaux rélatifs aux variétés de Stein" 26775:"Quelques problèmes globaux rélatifs aux variétés de Stein" 26334:

Japanese Journal of Mathematics: Transactions and Abstracts

25810:

First Steps in Several Complex Variables: Reinhardt Domains

24386: 23945:. The latter means that it has a strongly pseudoconvex (or 23715:{\displaystyle A=\{x\in X;f^{-1}f(x_{0})\ (v=1,\dots ,k)\}} 23170:

Levi's problem remains unresolved in the following cases;

23162:, a generalized in the singular case of complex manifolds. 17548:{\displaystyle Q(t):=\{Z_{j}=\varphi _{j}(u,t);|u|\leq 1\}} 17430:{\displaystyle {\frac {\partial \varphi _{j}}{\partial u}}} 13042:

The sets in the definition. Note: On this section, replace

5187:

Power series expansion of holomorphic functions on polydisc

1666:

is holomorphic in each variable separately. Conversely, if

621: 26106: 23941:

Being a Stein manifold is equivalent to being a (complex)

23344:

is a locally pseudoconvex map (i.e. Stein morphism). Then

21475:. The obstruction to defining a logarithm at the level of 18843:) helped solve the problems in several complex variables. 16322: 15500:, that can be second-order differentiable with respect to 15326:{\displaystyle f\colon X\to \mathbb {R} \cup \{-\infty \}} 14918: 10104:{\displaystyle a=(a_{1},\dots ,a_{n})\in \mathbb {C} ^{n}} 30267:

Theory of Analytic Functions of Several Complex Variables

24401:

in the sense of so-called "holomorphic homotopy theory".

23150:

is a holomorphically convex (i.e. Stein manifold). Also,

22195:

is holomorphically convex, i.e. for every compact subset

20902:

is surjective, then Second Cousin problem can be solved:

15921: 13493: 12649:(1978) established a generalization of Thullen's result: 11595:

is a logarithmically-convex Reinhardt domain with centre

8195:

Bochner–Martinelli formula (Cauchy's integral formula II)

2404:

Prove the sufficiency of two conditions (A) and (B). Let

30556:(Second ed.). AMS Chelsea Publishing. p. 340. 30464:

An Introduction to Complex Analysis in Several Variables

30182:

Theorie der Funktionen mehrerer komplexer Veränderlichen

29778:

Holomorphic Vector Bundles over Compact Complex Surfaces

27991:"A Weak Coherence Theorem and Remarks to the Oka Theory" 24869:

In fact, this was proved by Kiyoshi Oka with respect to

23040:

of the Stein manifold X is a Locally pseudoconvex, then

18965:

which has a local presentation, that is, every point in

15860:{\displaystyle {\sqrt {-1}}\partial {\bar {\partial }}f} 6610: 584:

formulation of the theory (with major repercussions for

31112:, Springer-Verlag Berlin Heidelberg, p. XIV, 226, 30526:

Complex Manifolds and Deformation of Complex Structures

25666:

Science Reports of the Tokyo Bunrika Daigaku, Section A

25050:{\displaystyle \mathbb {C} ^{n}\times \mathbb {P} _{m}} 24405:

Complex projective varieties (compact complex manifold)

24256:

agreeing with the usual orientation as the boundary of

23722:. This concept also applies to complex analytic space. 21998:

would restrict to nonconstant holomorphic functions on

19988:

of complex submanifold or every complex analytic space

18827:. Cartan translated this notion into the notion of the 18097:

with smooth boundary on non-Kähler manifolds such that

17650:{\displaystyle B(t):=\{Z_{j}=\varphi _{j}(u,t);|u|=1\}} 10185:{\displaystyle z^{0}=(z_{1}^{0},\dots ,z_{n}^{0})\in D} 481:

After 1945 important work in France, in the seminar of

31296:

This article incorporates material from the following

31285: 30582:. Graduate Texts in Mathematics. Vol. 108. 1986. 28463: 27919:

Remmert, R. (1994). "Local Theory of Complex Spaces".

27596:

Science Reports of the Tokyo Kyoiku Daigaku, Section A

26411:

Huckleberry, Alan (2013). "Hans Grauert (1930–2011)".

24374:

taking values in the complex numbers. See for example

24014:{\displaystyle i\partial {\bar {\partial }}\psi >0} 18856:

The definition of the coherent sheaf is as follows. A

16398:{\displaystyle \rho :\mathbb {C} ^{n}\to \mathbb {R} } 15335:

is said to be plurisubharmonic if and only if for any

13654: 13551: 11363: 6684: 6265:

which converges uniformly on compacta inside a domain

1096: 30606:

Introduction to Complex Analysis in Several Variables

29883:

Théorie des Intersections et Théorème de Riemann-Roch

27629: 27212: 26802: 26456:

Oka, Kiyoshi (1953). Merker, j.; Noguchi, j. (eds.).

26107:

Boggess, A.; Dwilewicz, R.J.; Slodkowski, Z. (2013).

25210: 25157: 25128: 25063: 25019: 24875: 24781: 24624: 24534: 24449: 24427: 24313: 24262: 24199: 24167:{\displaystyle \{z\mid -\infty \leq \psi (z)\leq c\}} 24124: 24096: 24027: 23983: 23955: 23903: 23816: 23769: 23737: 23625: 23598: 23556: 23527: 23477: 23444: 23366: 23314: 23268: 23239: 23210: 23184: 23114: 23070: 23020: 22948: 22850: 22677: 22606: 22526: 22473: 22434: 22404: 22233: 22201: 22149: 22129: 22072: 22012: 22002:, contradicting compactness, except in the case that 21975: 21946: 21909: 21877: 21839: 21768: 21742: 21692: 21532: 21485: 21450: 21374: 21327: 21298: 21247: 21188: 21034: 20911: 20888: 20842: 20809: 20780: 20742: 20718: 20676: 20522: 20416: 20374: 20347: 20310: 20286: 20246: 20219: 20192: 20108: 20072: 20039: 19994: 19963: 19934: 19880: 19836: 19812: 19699: 19657: 19633: 19602: 19523: 19497: 19471: 19448: 19375: 19355: 19335: 19312: 19281: 19253: 19226: 19183: 19156: 19136: 19017: 18991: 18971: 18936: 18912: 18869: 18801: 18729: 18657: 18596: 18559: 18515: 18491: 18471: 18439: 18413: 18360: 18327: 18321:) and was solved for unramified Riemann domains over 18293: 18267: 18241: 18209: 18131: 18103: 18083: 18054: 18015: 17976: 17950: 17918: 17884: 17843: 17806: 17763: 17716: 17663: 17565: 17555:

is called an analytic disc de-pending on a parameter

17463: 17443: 17399: 17359: 17299: 17238: 17197: 17177: 17153: 17129: 17102: 17055: 17035: 16975: 16936: 16903: 16857: 16690: 16577: 16554: 16525: 16480: 16442: 16411: 16368: 16333: 16296: 16233: 16146: 16075: 16007: 15983: 15932: 15889: 15874: 15828: 15793: 15665: 15642: 15611: 15591: 15510: 15471: 15440: 15382: 15347: 15290: 15265: 15199: 15151: 15106: 15042: 14991: 14932: 14858: 14799: 14766: 14737: 14717: 14687: 14630: 14582: 14538: 14440: 14406: 14371: 14331: 14311: 14285: 14232: 14208: 14168: 14004: 13967: 13930: 13906: 13886: 13851: 13816: 13769: 13742: 13715: 13641:{\displaystyle D_{1}\subseteq D_{2}\subseteq \cdots } 13608: 13505: 13456: 13430: 13397: 13349: 13323: 13288: 13262: 13229: 13165: 13118: 13079: 13048: 12989: 12960: 12927: 12868: 12848: 12767: 12717: 12690: 12663: 12505: 12395: 12290: 12245: 12218: 12191: 12164: 12137: 12110: 12031: 11822: 11785: 11673: 11640: 11601: 11555: 11334: 11302: 11184: 11057: 11018: 10797: 10761: 10601: 10565: 10445: 10386: 10201: 10117: 10044: 10018: 9983: 9933: 9909: 9870: 9831: 9807: 9776: 9702: 9676: 9652: 9602: 9565: 9526: 9487: 9458: 9400: 9356: 9314: 9282: 9253: 9182: 9153: 9091: 9036: 8989: 8898: 8897: 8752: 8751: 8457: 8416: 8396: 8357: 8328: 8304: 8272: 8249: 8220: 8154: 8111: 8049: 7438: 7268: 7236: 7050: 6876: 6621: 6349: 6310: 6279: 6225: 6007: 5971: 5892: 5374: 5201: 4906: 4865: 4691: 4653: 4179: 3878: 3838: 3097: 3046: 2847: 2773: 2746: 2669: 2601: 2574: 2547: 2503: 2472: 2441: 2421: 2365: 2267: 2086: 2066: 2046: 2026: 1983: 1832: 1812: 1788: 1768: 1740: 1704: 1555: 1450: 1405: 1379: 1357: 1322: 1276: 1247: 1231:

the (real) orientation of the space is never reversed

1158: 1090: 970: 938: 912: 857: 824: 795: 773: 734: 540: 507: 411: 319:, and also, as an example of an inverse problem; the 268: 236: 178: 146: 38: 29263:"The Behnke-Stein Theorem for Open Riemann Surfaces" 28382: 28380: 28378: 27473: 26413:

Jahresbericht der Deutschen Mathematiker-Vereinigung

25122:

This is the standard method for compactification of

24758:

Note that this formula only holds for polydisc. See

23897:

is holomorphically spreadable, i.e. for every point

13205:{\displaystyle D\subset \mathbb {C} ^{n}\ (n\geq 2)} 11630:

Hartogs's extension theorem and Hartogs's phenomenon

9864:. In other words, there may be not exist a function 474:

over a two-dimensional surface. This means that the

31078: 29085: 27871: 26981:Fritzsche, Klaus; Grauert, Hans (6 December 2012). 24939:

This relation is called the Cartan–Thullen theorem.

23949:) exhaustive function, i.e. a smooth real function 23404:{\displaystyle D=\bigcup _{n\in \mathbb {N} }D_{n}} 23007:Cartan extended Levi's problem to Stein manifolds. 18785:{\displaystyle (f+f',\delta \cap \delta ')\in (I).} 18228:{\displaystyle 1\Leftrightarrow 2\Leftrightarrow 3} 16206: 15504:of one-variable complex function is subharmonic is 13545:are domains of holomorphy, then their intersection 13143:and the boundary is called the natural boundary of 9424:biholomorphically equivalent, that is, there is no 1523:{\displaystyle f(z+h)=f(z)+L(h)+o(\lVert h\rVert )} 563:and Stein manifolds, satisfying a condition called 397:. Hartogs proved some basic results, such as every 215:{\displaystyle D\subset \mathbb {C} ^{n},\ n\geq 2} 30665: 27644: 27220: 26406: 26404: 26402: 26400: 26367: 25225: 25165: 25143: 25078: 25049: 24890: 24794: 24642: 24552: 24484: 24435: 24347: 24299: 24237: 24193:, the field of complex tangencies to the preimage 24166: 24102: 24066: 24013: 23961: 23915: 23840: 23784: 23752: 23714: 23611: 23584: 23542: 23509: 23463: 23403: 23332: 23280: 23254: 23225: 23196: 23126: 23085: 23032: 22984: 22934: 22829: 22656: 22545: 22506: 22459: 22416: 22370: 22213: 22168: 22135: 22095:is connection to a manifold, can be regarded as a 22087: 22066:in algebraic geometry. If the univalent domain on 22027: 21990: 21961: 21924: 21895: 21847: 21804: 21754: 21728: 21671: 21515: 21467: 21433: 21357: 21313: 21284: 21231: 21171: 21014: 20894: 20874: 20824: 20795: 20763: 20724: 20704: 20631: 20498: 20380: 20360: 20329: 20292: 20272: 20240:where they are defined, and where each difference 20232: 20205: 20125: 20087: 20058: 20023: 19980: 19949: 19916: 19853: 19822: 19798: 19674: 19639: 19619: 19588: 19509: 19483: 19454: 19434: 19361: 19341: 19318: 19298: 19263: 19236: 19212: 19162: 19142: 19119: 18997: 18977: 18953: 18922: 18898: 18819: 18784: 18715: 18640: 18582: 18545: 18497: 18477: 18457: 18425: 18375: 18342: 18305: 18279: 18253: 18227: 18152: 18109: 18089: 18069: 18032: 18001: 17962: 17936: 17896: 17858: 17821: 17784: 17737: 17702: 17649: 17547: 17449: 17429: 17381: 17345: 17285: 17203: 17183: 17159: 17139: 17108: 17072: 17041: 17018: 16953: 16922: 16874: 16836: 16674: 16560: 16540: 16507: 16466: 16428: 16397: 16350: 16305: 16282: 16195: 16124: 16049: 15989: 15955: 15906: 15859: 15810: 15776: 15648: 15628: 15597: 15577: 15492: 15454: 15423: 15365: 15325: 15271: 15243: 15181: 15133: 15092: 15014: 14972: 14891: 14844: 14785: 14752: 14723: 14695: 14657: 14616: 14568: 14516: 14418: 14389: 14353: 14317: 14297: 14260: 14214: 14187: 14147: 13979: 13949: 13912: 13892: 13872: 13828: 13795: 13755: 13728: 13694: 13640: 13591: 13537: 13470: 13442: 13409: 13367: 13335: 13309: 13274: 13244: 13204: 13127: 13108:and cannot directly connect to the domain outside 13100: 13054: 13004: 12975: 12942: 12903: 12854: 12834: 12753: 12703: 12676: 12629: 12489: 12379: 12258: 12231: 12204: 12177: 12150: 12123: 12046: 11986: 11803: 11771: 11655: 11613: 11587: 11541: 11349: 11317: 11281: 11164: 11040: 10958: 10780: 10736: 10584: 10548: 10427: 10362: 10184: 10103: 10030: 10004: 9945: 9918: 9895: 9856: 9813: 9785: 9738: 9688: 9658: 9638: 9584: 9551: 9512: 9473: 9412: 9362: 9342: 9300: 9268: 9201: 9168: 9139: 9077: 9010: 8963: 8873: 8736: 8440: 8402: 8378: 8343: 8310: 8278: 8258: 8235: 8183: 8140: 8097: 8032: 7421: 7251: 7210: 7036: 6862: 6670: 6599: 6323: 6292: 6257: 6203: 5986: 5957: 5868: 5357: 5168: 4889: 4851: 4670: 4620: 4133: 3847: 3821: 3076: 3032: 2833: 2759: 2729: 2655: 2587: 2560: 2533: 2489: 2454: 2427: 2389: 2347: 2247: 2072: 2052: 2032: 2013:{\displaystyle f:\mathbb {C} ^{n}\to \mathbb {C} } 2012: 1967: 1818: 1794: 1774: 1754: 1726: 1652: 1522: 1435:{\displaystyle L:\mathbb {C} ^{n}\to \mathbb {C} } 1434: 1391: 1365: 1343: 1291: 1262: 1210: 1135: 988: 953: 920: 872: 839: 810: 781: 749: 555: 515: 441:{\displaystyle f:\mathbb {C} ^{n}\to \mathbb {C} } 440: 283: 254: 214: 160: 53: 30785: 29092:Transactions of the American Mathematical Society 28428: 28375: 26113:Journal of Mathematical Analysis and Applications 25696:Several Complex Variables and Complex Manifolds I 25609:"Vanishing theorems on complete Kähler manifolds" 24732:A name adopted, confusingly, for the geometry of 22657:{\displaystyle H^{1}(X,{\mathcal {O}}_{X}^{*})=0} 20875:{\displaystyle \mathbf {K} ^{*}/\mathbf {O} ^{*}} 19861:are coherent, then the third is coherent as well. 16362:has a defining function; i.e., that there exists 16050:{\displaystyle \{z\in X;\varphi (z)\leq \sup x\}} 14617:{\displaystyle D^{c}=\mathbb {C} ^{n}\setminus D} 10428:{\displaystyle \theta _{\nu }\;(\nu =1,\dots ,n)} 4897:, the following evaluation equation is obtained. 1670:is holomorphic in each variable separately, then 1547:is holomorphic, then all the partial maps : 30: 31342: 31302:Creative Commons Attribution/Share-Alike License 29402:. Graduate Studies in Mathematics. Vol. 5. 29268:Proceedings of the American Mathematical Society 29230:"Some remarks on parallelizable Stein manifolds" 29161: 28608:"Another Direct Proof of Oka's Theorem (Oka IX)" 27288: 27103: 27101: 27099: 27097: 26980: 26667: 26540: 26538: 26020:Proceedings of the American Mathematical Society 24389:set of analogies, Stein manifolds correspond to 22292: 20133:is a coherent sheaf, also, (i) is used to prove 19865: 18716:{\displaystyle (f,\delta ),(f',\delta ')\in (I)} 18641:{\displaystyle (af,\delta \cap \delta ')\in (I)} 18044:is locally pseudoconvex (or locally Stein) over 17286:{\displaystyle \varphi :z_{j}=\varphi _{j}(u,t)} 17019:{\displaystyle D=\bigcup _{k=1}^{\infty }D_{k}.} 16038: 15244:{\displaystyle \{z\in \mathbb {C} ;a+bz\in D\}.} 14070: 9639:{\displaystyle U,\ V,\ U\cap V\neq \varnothing } 717:theory, that draw on several complex variables. 645:theory of functions of several complex variables 30484:Analytic Functions of Several Complex Variables 30285: 29815:Journal für die reine und angewandte Mathematik 28986: 28984: 28982: 28980: 28867: 28865: 28863: 28861: 27872:Grothendiec, Alexander; Dieudonn, Jean (1960). 27142: 27140: 26984:From Holomorphic Functions to Complex Manifolds 26489: 26487: 26485: 26397: 26109:"Hartogs extension for generalized tubes in Cn" 25412:"Complex analysis in one and several variables" 25197: 25195: 20145:In the case of one variable complex functions, 20059:{\displaystyle {\mathcal {I}}\langle A\rangle } 17346:{\displaystyle \Delta :|U|\leq 1,0\leq t\leq 1} 12862:being a permutation of the indices), such that 6671:{\displaystyle \{r_{\nu }\ (\nu =1,\dots ,n)\}} 1734:defined on the plane is holomorphic at a point 1674:is in fact holomorphic : this is known as 720: 659:, certainly, is in specific generalizations of 30688: 30481:Gunning, Robert Clifford; Rossi, Hugo (2009). 30457: 30386: 29086:Andreotti, Aldo; Narasimhan, Raghavan (1964). 28780: 28778: 28288:Journal of Science of the Hiroshima University 28255:Journal of Science of the Hiroshima University 28193:Journal of Science of the Hiroshima University 28111: 28066: 28039: 27679: 27677: 26691: 26689: 26687: 26685: 26683: 26529:Iwanami Shoten, Tokyo (Oka's Original Version) 26327: 26325: 26323: 26282: 26280: 25987: 25985: 25951: 25949: 25689: 25687: 25550:"Géométrie algébrique et géométrie analytique" 25095:The proof method uses an approximation by the 24771:According to the Jordan curve theorem, domain 23890:for Riemann surfaces, due to Behnke and Stein. 20155:analytic function in several complex variables 13957:stand for the set of holomorphic functions on 13840: 13490:for a domain of definition of its reciprocal. 12082:is connected, then every holomorphic function 10981:. Therefore, the complete Reinhardt domain is 4157: 2400:Cauchy's integral formula I (Polydisc version) 2355:or even more compactly using the formalism of 1727:{\displaystyle f:\mathbb {C} \to \mathbb {C} } 713:. There are a number of other fields, such as 478:will have to take a very different character. 31186:Bulletin of the American Mathematical Society 31159:Bulletin of the American Mathematical Society 31132:Bulletin of the American Mathematical Society 30695: 30206: 30179: 30024: 29920:Bulletin de la Société Mathématique de France 29234:Bulletin of the American Mathematical Society 29011: 28942: 28695:Bulletin de la Société Mathématique de France 27517: 27373:"Weakly 1-Complete Manifold and Levi Problem" 27257: 27255: 27107: 27094: 27041:Bulletin de la Société Mathématique de France 26632: 26630: 26607:Bulletin de la Société Mathématique de France 26596: 26594: 26592: 26590: 26535: 26500:Bulletin de la Société Mathématique de France 26250: 25806: 25770:"Uniformization in Several Complex Variables" 25366:Bulletin of the American Mathematical Society 24067:{\displaystyle \{z\in X\mid \psi (z)\leq c\}} 23411:an increasing union of Stein open sets. Then 20186:, one is given several meromorphic functions 20066:of an analytic subset A of an open subset of 18121:Conditions equivalent to domain of holomorphy 14430:satisfies the above holomorphic convexity on 13112:, including the point of the domain boundary 8888:is holomorphic the second term vanishes, so 8098:{\displaystyle r'_{\nu }<|z|<R'_{\nu }} 2080:satisfiy the Cauchy Riemann equations : 86:complex analysis of functions of one variable 30554:Function Theory of Several Complex Variables 30480: 30140:Journal of the Mathematical Society of Japan 28977: 28907: 28871: 28858: 28823: 28336: 28334: 28096:"Basic results on Sheaves and Analytic Sets" 28067:Grauert, H.; Remmert, R. (6 December 2012). 28040:Grauert, H.; Remmert, R. (6 December 2012). 27815: 27813: 27811: 27593: 27289:Diederich, Klas; Fornæss, John Erik (1982). 27137: 27110:Journal of the Mathematical Society of Japan 27011:Function Theory of Several Complex Variables 26936:Notices of the American Mathematical Society 26805:"Séminaire Henri Cartan, Tome 4 (1951-1952)" 26670:"Séminaire Henri Cartan, Tome 3 (1950-1951)" 26568:Journal of the Mathematical Society of Japan 26547:Journal of the Mathematical Society of Japan 26482: 26143:Journal de Mathématiques Pures et Appliquées 26055: 25733:Rendiconti del Circolo Matematico di Palermo 25297: 25295: 25204:"L estimates and existence theorems for the 25192: 24812:a variable is other than 0, it may converge. 24775:is bounded closed set, that is, each domain 24189:such that, away from the critical points of 24161: 24125: 24061: 24028: 23867:Every Stein manifold of (complex) dimension 23709: 23632: 23471:, there exist finitely many holomorphic map 22176:denote the ring of holomorphic functions on 21821:Stein manifold (non-compact Kähler manifold) 21232:{\displaystyle H^{1}(M,\mathbf {O} ^{*})=0.} 20053: 20047: 17644: 17581: 17542: 17479: 17049:as in the definition we can actually find a 16568:in the complex tangent space at p, that is, 16502: 16490: 16461: 16449: 16044: 16008: 15418: 15409: 15320: 15311: 15235: 15200: 15072: 15043: 14964: 14955: 13695:{\textstyle D=\bigcup _{n=1}^{\infty }D_{n}} 13592:{\textstyle D=\bigcap _{\nu =1}^{n}D_{\nu }} 12599: 12506: 12484: 12396: 12374: 12291: 12269: 11957: 11836: 11766: 11687: 11159: 11071: 9592:is the set/ring of holomorphic functions on 6665: 6622: 5352: 5202: 3855:is a rectifiable Jordanian closed curve and 3054: 3047: 2295: 2277: 2114: 2096: 1514: 1508: 30821: 30749: 30731: 30713: 30502:Kaup, Ludger; Kaup, Burchard (9 May 2011). 29775: 29744: 29434:Algebraic Geometry over the Complex Numbers 29012:Fornæss, John Erik; Sibony, Nessim (2001). 28775: 27674: 27179: 26861: 26680: 26668:Cartan, H.; Eilenberg, Samuel; Serre, J-P. 26410: 26320: 26277: 25982: 25946: 25684: 25639: 23340:a Riemann unbranched domain, such that map 20658:, the Cousin problem is always solvable if 17908:cannot be extended to any neighbourhood of 15956:{\displaystyle X\subset {\mathbb {C} }^{n}} 15926:Weak pseudoconvex is defined as : Let 15455:{\displaystyle \Delta \subset \mathbb {C} } 15366:{\displaystyle \varphi \colon \Delta \to X} 15015:{\displaystyle D\subset {\mathbb {C} }^{n}} 14658:{\displaystyle (D\subset \mathbb {C} ^{n})} 13147:. In other words, the domain of holomorphy 8191:, and so it is possible to integrate term. 7225: 6300:also uniformly on compacta inside a domain 1693: 1233:by a complex operator. The same applies to 343:, that addresses the generalization of the 31245:Complex Analytic and Differential Geometry 30992: 30359: 29914:Borel, Armand; Serre, Jean-Pierre (1958). 29913: 29847: 29745:Cartan, Henri; Serre, Jean-Pierre (1953). 29492: 28948: 28913: 28754: 27252: 26970:Complex Analytic and Differential Geometry 26824: 26627: 26587: 25471: 25405: 25403: 25401: 25399: 25397: 24978:This is called the classic Cousin problem. 23863:(because the embedding is biholomorphic). 23726:Properties and examples of Stein manifolds 23158:And Narasimhan extended Levi's problem to 21523:, from the long exact cohomology sequence 21321:can be compared with the cohomology group 20399:be the sheaf of meromorphic functions and 18179:is the union of an increasing sequence of 17091: 16961:-boundary which are relatively compact in 14365:with the relatively compact components of 10397: 709:, both of which had some inspiration from 133: 31209:"From Riemann Surfaces to Complex Spaces" 31197: 31170: 31143: 30947: 30875: 30857: 30803: 30786:Gonchar, A.A.; Vladimirov, V.S. (2001) , 30387:Grauert, Hans; Remmert, Reinhold (1979), 30151: 30118: 30100: 30077: 29931: 29880: 29850:Topological Methods in Algebraic Geometry 29672: 29654: 29345: 29303: 29280: 29245: 29103: 28996: 28798: 28739: 28706: 28626: 28532: 28477: 28331: 28299: 28266: 28204: 28182: 28180: 28178: 28166: 28009: 27808: 27697: 27632: 27495: 27449: 27404:Heinrich Behnke & Karl Stein (1951), 27388: 27347: 27243: 27214: 27164: 27121: 27085: 27052: 26947: 26914: 26618: 26579: 26558: 26511: 26424: 26381: 26345: 26268: 26227: 26186: 26124: 26083: 26073: 26032: 26015:"Some remarks about a theorem of Hartogs" 25997: 25859: 25744: 25624: 25567: 25427: 25377: 25362:"Pseudoconvexity and the problem of Levi" 25355: 25353: 25351: 25349: 25347: 25345: 25343: 25341: 25339: 25319: 25292: 25248: 25201: 25159: 25131: 25066: 25037: 25022: 24878: 24822: 24630: 24627: 24572:. As an application of this theorem, the 24540: 24537: 24472: 24469: 24454: 24429: 23819: 23772: 23740: 23530: 23385: 23073: 22969: 22781: 22075: 22015: 21978: 21949: 21912: 21880: 21841: 21789: 21629: 21506: 21461: 21391: 21285:{\displaystyle H^{1}(M,\mathbf {O} ^{*})} 20075: 19937: 19902: 19369:such that there is a surjective morphism 19244:satisfying the following two properties: 18330: 18160:the following conditions are equivalent: 18153:{\displaystyle D\subset \mathbb {C} ^{n}} 18140: 18057: 17809: 16773: 16391: 16377: 15942: 15748: 15548: 15448: 15402: 15304: 15210: 15121: 15080: 15068: 15001: 14947: 14740: 14689: 14642: 14598: 13873:{\displaystyle G\subset \mathbb {C} ^{n}} 13860: 13464: 13375:such that for every holomorphic function 13310:{\displaystyle V\subset \mathbb {C} ^{n}} 13297: 13232: 13174: 13101:{\displaystyle D\subset \mathbb {C} ^{n}} 13088: 12992: 12963: 12930: 12741: 12726: 12602: 12529: 12419: 12314: 12034: 11698: 11643: 11337: 11305: 10996: 10091: 10038:) to be a domain, with centre at a point 10005:{\displaystyle D\subset \mathbb {C} ^{n}} 9992: 9461: 9390:The Riemann mapping theorem does not hold 9256: 9156: 9011:{\displaystyle D\subset \mathbb {C} ^{n}} 8998: 8630: 8331: 8223: 7333: 7115: 6941: 6304:. Also, respective partial derivative of 5945: 5832: 5264: 4756: 4585: 4101: 3801: 3618: 3581: 3506: 3342: 3267: 2940: 2006: 1992: 1748: 1720: 1712: 1428: 1414: 1359: 1344:{\displaystyle D\subset \mathbb {C} ^{n}} 1331: 1279: 1250: 973: 941: 914: 860: 827: 798: 775: 737: 581:Grauert–Riemenschneider vanishing theorem 543: 509: 434: 420: 302: 271: 242: 239: 187: 154: 41: 31254: 31108:Oka, Kiyoshi (1984), Remmert, R. (ed.), 30501: 30362:Stein Manifolds and Holomorphic Mappings 30286:Cartan, Henri; Takahashi, Reiji (1992). 30133: 29126: 27146: 26965: 26963: 26961: 26959: 26896: 26831:Stein Manifolds and Holomorphic Mappings 25920:Complex Variables and Elliptic Equations 25726: 25663: 23927:holomorphic functions defined on all of 23093:, and reducing it to the result of Oka. 23044:is a Stein manifold, and conversely, if 22577:of Behnke and Stein (1948) asserts that 21829:, the open Riemann surface is in fact a 21805:{\displaystyle H^{2}(M,\mathbb {Z} )=0.} 20665: 18033:{\displaystyle {\mathcal {C}}^{\infty }} 17073:{\displaystyle {\mathcal {C}}^{\infty }} 16954:{\displaystyle {\mathcal {C}}^{\infty }} 14272:. Sometimes this is just abbreviated as 13037: 12000:Hartogs's extension theorem (1906); Let 11667:On the polydisk consisting of two disks 9443: 4671:{\displaystyle {\mathcal {C}}^{\infty }} 1302: 605:deformation theory of complex structures 31300:articles, which are licensed under the 31239:Tasty Bits of Several Complex Variables 31206: 31179: 30965: 30893: 30839: 30630: 30522: 30337:. Universitext (2 ed.). Springer. 30327: 30225: 29954: 29628: 29593: 29558: 29457: 29430: 29395: 29227: 29196: 28990: 28829: 28649: 28605: 27988: 27982: 27918: 27771: 27728: 27683: 27620: 27034: 26695: 26168: 25991: 25955: 25506: 25394: 22995:This is related to the solution of the 22985:{\displaystyle H^{2}(X,\mathbb {Z} )=0} 22668:leads to the following exact sequence: 21812:(This condition called Oka principle.) 21729:{\displaystyle H^{q}(M,\mathbf {O} )=0} 20392:behaviour of the given local function. 20177: 17703:{\displaystyle Q(t)\subset D\ (0<t)} 17223: 16323:(Weakly) Levi(–Krzoska) pseudoconvexity 15656:is plurisubharmonic if and only if the 15134:{\displaystyle a,b\in \mathbb {C} ^{n}} 14919:Definition of plurisubharmonic function 13256:if there do not exist non-empty domain 13033: 12657:-dimensional bounded Reinhardt domains 9758:is said to be analytic continuation of 3084:be the center of each disk.) Using the 31343: 31125: 31050: 30929: 30911: 30767: 30551: 30523:Kodaira, Kunihiko (17 November 2004). 30304: 30059: 29360: 29327: 29057:. Springer Monographs in Mathematics. 29050: 28784: 28688: 28175: 28144: 27960:. Springer Monographs in Mathematics. 27953: 27370: 27327: 27261: 27203: 27007: 26798: 26796: 26636: 26600: 26251:Cartan, Henri; Thullen, Peter (1932). 26209: 26139: 26012: 25917: 25807:Jarnicki, Marek; Pflug, Peter (2008). 25694:Field, M (1982). "Complex Manifolds". 25606: 25540: 25538: 25536: 25336: 25301: 25271: 24082:. This is a solution to the so-called 22997:second (multiplicative) Cousin problem 22460:{\displaystyle f\in {\mathcal {O}}(X)} 20882:is well-defined. If the following map 20403:the sheaf of holomorphic functions on 20024:{\displaystyle (X,{\mathcal {O}}_{X})} 19213:{\displaystyle (X,{\mathcal {O}}_{X})} 18899:{\displaystyle (X,{\mathcal {O}}_{X})} 17904:be holomorphically convex.) such that 16508:{\displaystyle \partial D=\{\rho =0\}} 15922:(Weakly) pseudoconvex (p-pseudoconvex) 14569:{\displaystyle {\text{dist}}(K,D^{c})} 13494:Properties of the domain of holomorphy 13223:-dimensional complex coordinate space 13139:is called the domain of holomorphy of 11804:{\displaystyle 0<\varepsilon <1} 10435:is a arbitrary real numbers, a domain 10111:, such that, together with each point 9896:{\displaystyle f\in {\mathcal {O}}(U)} 9857:{\displaystyle g\in {\mathcal {O}}(V)} 9801:, such that all holomorphic functions 9552:{\displaystyle g\in {\mathcal {O}}(V)} 9513:{\displaystyle f\in {\mathcal {O}}(U)} 9147:coincide on the whole complex line of 2390:{\displaystyle {\bar {\partial }}f=0.} 588:, in particular from Grauert's work). 579:groups vanish, on the other hand, the 23:functions of several complex variables 31152: 30696:Gonchar, A.A.; Shabat, B.V. (2001) , 30529:. Classics in Mathematics. Springer. 30412: 29695: 29399:Algebraic Curves and Riemann Surfaces 29260: 28721: 28563: 28518: 28219: 27819: 26956: 26929: 26772: 26698:Archive for History of Exact Sciences 26476:"Related to Works of Dr. Kiyoshi OKA" 26056:Merker, Joël; Porten, Egmont (2007). 25882: 25841: 25693: 25544: 25467: 25465: 25463: 25275:Analysis of Several Complex Variables 24707: 21516:{\displaystyle H^{2}(M,\mathbb {Z} )} 21358:{\displaystyle H^{1}(M,\mathbf {O} )} 17944:be a holomorphic map, if every point 16897:, strongly Levi pseudoconvex domains 16227:plurisubharmonic exhaustion function 14390:{\displaystyle G\setminus K\subset G} 14325:is holomorphically convex since then 13471:{\displaystyle D\subset \mathbb {C} } 12904:{\displaystyle \varphi (G_{1})=G_{2}} 11663:were all connected to larger domain. 9276:, the bijective holomorphic function 9018:and which coincide on an open subset 8184:{\displaystyle R'_{\nu }<R_{\nu }} 8141:{\displaystyle r'_{\nu }>r_{\nu }} 6611:Radius of convergence of power series 2760:{\displaystyle {\overline {\Delta }}} 2261:, this can be reformulated as : 1682:under the additional hypothesis that 1373:is said to be holomorphic at a point 701:Subsequent developments included the 671:. These days these are associated to 629:pinned down the crossover point from 161:{\displaystyle D\subset \mathbb {C} } 30311:. Courier Corporation. p. 228. 30264: 30247: 29989: 29808: 28340: 28093: 27878:Publications Mathématiques de l'IHÉS 27067: 25409: 21292:for the multiplicative structure on 20102:From the above Serre(1955) theorem, 18485:holomorphic on a non-empty open set 15918:a strict plurisubharmonic function. 15189:is a subharmonic function on the set 13702:is also a domain of holomorphy (see 13486:of the domain, which must then be a 13383:there exists a holomorphic function 9428:between the two. This was proven by 9420:, open balls and open polydiscs are 9140:{\displaystyle g(z_{1},z_{2})=z_{1}} 4170: 3869: 1698:In one complex variable, a function 1223:composed from 2 × 2 blocks 1211:{\displaystyle u^{2}+v^{2}=|w|^{2}.} 1048:Any such space, as a real space, is 851:, and more generalized Stein space. 565: 339:; it did justify the local picture, 31282:Topics in Several Complex Variables 31107: 28281: 28248: 28186: 26793: 26661: 26565: 26544: 26519: 26493: 26473: 26455: 26331: 26286: 25767: 25533: 25359: 25185: 24300:{\displaystyle f^{-1}(-\infty ,c).} 23841:{\displaystyle \mathbb {C} ^{2n+1}} 23096:Also, Grauert proved for arbitrary 23056:is a Stein manifold if and only if 22592:(1956), states moreover that every 22569:be a connected, non-compact (open) 21969:, then the coordinate functions of 17792:holds on any family of Oka's disk, 17218: 16221:Strongly (or Strictly) pseudoconvex 15822:is plurisubharmonic if and only if 14845:{\displaystyle |z_{2}|<R(z_{1})} 14576:denotes the distance between K and 12641: 10192:, the domain also contains the set 9968: 9350:is also holomorphic. At this time, 8974: 7044:and does not converge uniformly at 6335:to the corresponding derivative of 6331:also compactly converges on domain 6258:{\displaystyle f_{1},\ldots ,f_{n}} 663:. The classical candidates are the 643:was heard to complain that the new 13: 30985: 30806:"Pseudo-convex and pseudo-concave" 30207:Bochner, S.; Martin, W.T. (1948). 29431:Arapura, Donu (15 February 2012). 29051:Ohsawa, Takeo (10 December 2018). 28724:"Families of nonnegative divisors" 27954:Ohsawa, Takeo (10 December 2018). 26932:"WHAT IS...a Pseudoconvex Domain?" 25511:. Universitext. pp. 300–346. 25460: 25304:"Cauchy–Riemann meet Monge–Ampère" 25226:{\displaystyle {\bar {\partial }}} 25214: 24528:into the complex projective space 24348:{\displaystyle f^{-1}(-\infty ,c)} 24333: 24282: 24181:with a real-valued Morse function 24137: 23993: 23987: 23510:{\displaystyle f_{1},\dots ,f_{k}} 23217: 23197:{\displaystyle D\subset \subset X} 22912: 22873: 22813: 22739: 22700: 22629: 22529: 22443: 22346: 22335: 22188:if the following conditions hold: 22152: 21468:{\displaystyle 2\pi i\mathbb {Z} } 20771:is holomorphic and non-vanishing. 20126:{\displaystyle {\mathcal {O}}^{p}} 20112: 20042: 20007: 19981:{\displaystyle {\mathcal {O}}_{X}} 19967: 19894: 19883: 19854:{\displaystyle {\mathcal {F}}_{j}} 19840: 19830:-modules two of the three sheaves 19815: 19767: 19738: 19709: 19675:{\displaystyle {\mathcal {O}}_{X}} 19661: 19620:{\displaystyle {\mathcal {O}}_{X}} 19606: 19569: 19533: 19415: 19379: 19299:{\displaystyle {\mathcal {O}}_{X}} 19285: 19256: 19229: 19196: 19130:for some (possibly infinite) sets 19094: 19058: 19021: 18954:{\displaystyle {\mathcal {O}}_{X}} 18940: 18915: 18882: 18546:{\displaystyle (f,\delta )\in (I)} 18376:{\displaystyle {\bar {\partial }}} 18364: 18104: 18084: 18025: 18019: 17850: 17444: 17418: 17403: 17300: 17132: 17065: 17059: 16998: 16946: 16940: 16893:is pseudoconvex, then there exist 16875:{\displaystyle {\mathcal {C}}^{2}} 16861: 16774: 16760: 16737: 16640: 16623: 16578: 16532: 16481: 16429:{\displaystyle {\mathcal {C}}^{2}} 16415: 16351:{\displaystyle {\mathcal {C}}^{2}} 16337: 16266: 16260: 16179: 16173: 16108: 16102: 15907:{\displaystyle {\mathcal {C}}^{2}} 15893: 15875:Strictly plurisubharmonic function 15845: 15839: 15811:{\displaystyle {\mathcal {C}}^{2}} 15797: 15749: 15735: 15721: 15629:{\displaystyle {\mathcal {C}}^{2}} 15615: 15549: 15542: 15528: 15511: 15441: 15415: 15395: 15354: 15317: 14961: 14718: 14668: 14171: 14158:One obtains a narrower concept of 14120: 13933: 13677: 13538:{\displaystyle D_{1},\dots ,D_{n}} 13119: 13049: 12220: 12139: 11878: 11675: 11588:{\displaystyle z_{1},\dots ,z_{n}} 11406: 9910: 9879: 9840: 9777: 9568: 9535: 9496: 9026:, are equal on the whole open set 8919: 8826: 8773: 8250: 7712: 7478: 6727: 6565: 6533: 6486: 6477: 6426: 6394: 6354: 6065: 5670: 5647: 5437: 5035: 5005: 4915: 4663: 4657: 4423: 4400: 4304: 4272: 4184: 3979: 3956: 3839: 3637: 3603: 3566: 3361: 3327: 3170: 3077:{\displaystyle \{z\}_{\nu =1}^{n}} 2850: 2776: 2749: 2534:{\displaystyle \nu =1,2,\ldots ,n} 2490:{\displaystyle {\mathcal {C}}^{1}} 2476: 2369: 2313: 2305: 2268: 2229: 2221: 2196: 2188: 2162: 2154: 2132: 2124: 2087: 1947: 1939: 1912: 1904: 1876: 1868: 1844: 1836: 611:was described in general terms by 136:. For one complex variable, every 79:Mathematics Subject Classification 29:dealing with functions defined on 14: 31367: 31232: 31081:The American Mathematical Monthly 31053:The American Mathematical Monthly 30269:. American Mathematical Society. 29705:Commentarii Mathematici Helvetici 29282:10.1090/S0002-9939-1989-0953748-X 29105:10.1090/S0002-9947-1964-0159961-3 28741:10.1090/S0002-9947-1968-0219751-3 28314: 27825:"Faisceaux algébriques cohérents" 27554:Commentarii Mathematici Helvetici 26897:Kajiwara, Joji (1 January 1965). 26034:10.1090/S0002-9939-1966-0201675-2 25500: 25308:Bulletin of Mathematical Sciences 24643:{\displaystyle \mathbb {CP} ^{n}} 24553:{\displaystyle \mathbb {CP} ^{n}} 24118:. A Stein domain is the preimage 22546:{\displaystyle {\mathcal {O}}(X)} 22169:{\displaystyle {\mathcal {O}}(X)} 21896:{\displaystyle \mathbb {R} ^{2n}} 20151:Weierstrass factorization theorem 20140: 18846: 17437:are not all zero at any point on 14608: 14434:it has the following properties. 14375: 14188:{\displaystyle {\mathcal {O}}(G)} 13950:{\displaystyle {\mathcal {O}}(G)} 13796:{\displaystyle D_{1}\times D_{2}} 10439:is invariant under the rotation: 9766:exists, for each way of choosing 9633: 9585:{\displaystyle {\mathcal {O}}(U)} 9378:biholomorphism also, we say that 9343:{\displaystyle \phi ^{-1}:V\to U} 9227: 8379:{\displaystyle \omega (\zeta ,z)} 2568:be the domain surrounded by each 1977:In several variables, a function 1755:{\displaystyle p\in \mathbb {C} } 989:{\displaystyle \mathbb {R} ^{2n}} 255:{\displaystyle \mathbb {CP} ^{n}} 116:. Equivalently, they are locally 30180:Behnke, H.; Thullen, P. (1934). 30127: 30094: 30053: 30018: 29983: 29948: 29907: 29874: 29841: 29802: 29769: 29738: 29689: 29622: 29587: 29552: 29486: 29451: 29424: 29389: 29354: 29321: 29297: 29254: 29221: 29190: 29155: 29120: 29044: 29005: 28748: 28715: 28682: 28643: 28599: 28557: 28512: 28457: 28422: 28347:The concept of a Riemann surface 28308: 28275: 28242: 27645:{\displaystyle \mathbb {C} ^{n}} 26013:Laufer, Henry B. (1 June 1966). 25416:Taiwanese Journal of Mathematics 25144:{\displaystyle \mathbb {C} ^{n}} 25116: 25106: 25089: 25079:{\displaystyle \mathbb {P} _{m}} 25008: 24999: 24990: 24981: 24972: 24920:Definition of weakly 1-complete. 24891:{\displaystyle \mathbb {C} ^{n}} 23785:{\displaystyle \mathbb {C} ^{n}} 23753:{\displaystyle \mathbb {C} ^{n}} 23619:is an isolated point of the set 23543:{\displaystyle \mathbb {C} ^{p}} 23438:is K-complete if, to each point 23134:of a arbitrary complex manifold 23086:{\displaystyle \mathbb {C} ^{n}} 23048:is a Locally pseudoconvex, then 23002: 22088:{\displaystyle \mathbb {C} ^{n}} 22028:{\displaystyle \mathbb {C} ^{n}} 21991:{\displaystyle \mathbb {C} ^{n}} 21962:{\displaystyle \mathbb {C} ^{n}} 21925:{\displaystyle \mathbb {C} ^{n}} 21713: 21659: 21584: 21553: 21415: 21399: 21348: 21314:{\displaystyle \mathbf {O} ^{*}} 21301: 21269: 21210: 21156: 21119: 21102: 21056: 20996: 20979: 20933: 20862: 20845: 20825:{\displaystyle \mathbf {K} ^{*}} 20812: 20796:{\displaystyle \mathbf {O} ^{*}} 20783: 20622: 20592: 20582: 20543: 20486: 20476: 20437: 20088:{\displaystyle \mathbb {C} _{n}} 19950:{\displaystyle \mathbb {C} _{n}} 18343:{\displaystyle \mathbb {C} ^{n}} 18070:{\displaystyle \mathbb {C} ^{n}} 17822:{\displaystyle \mathbb {C} ^{n}} 17749:is called Family of Oka's disk. 16467:{\displaystyle D=\{\rho <0\}} 16358:boundary , it can be shown that 16207:Strongly (Strictly) pseudoconvex 15182:{\displaystyle z\mapsto f(a+bz)} 14892:{\displaystyle -\log {R}(z_{1})} 14852:is a domain of holomorphy. Then 14753:{\displaystyle \mathbb {C} ^{2}} 14261:{\displaystyle K,{\hat {K}}_{G}} 13763:are domains of holomorphy, then 13368:{\displaystyle U\subset D\cap V} 13245:{\displaystyle \mathbb {C} ^{n}} 13005:{\displaystyle \mathbb {C} ^{n}} 12976:{\displaystyle \mathbb {C} ^{2}} 12943:{\displaystyle \mathbb {C} ^{n}} 12259:{\displaystyle H_{\varepsilon }} 12205:{\displaystyle H_{\varepsilon }} 12178:{\displaystyle H_{\varepsilon }} 12124:{\displaystyle H_{\varepsilon }} 12047:{\displaystyle \mathbb {C} ^{n}} 11656:{\displaystyle \mathbb {C} ^{n}} 11350:{\displaystyle \mathbb {C} ^{n}} 11318:{\displaystyle \mathbb {R} ^{n}} 9825:, have an analytic continuation 9762:. From the identity theorem, if 9474:{\displaystyle \mathbb {C} ^{n}} 9269:{\displaystyle \mathbb {C} ^{n}} 9169:{\displaystyle \mathbb {C} ^{2}} 9078:{\displaystyle f(z_{1},z_{2})=0} 8344:{\displaystyle \mathbb {C} ^{n}} 8236:{\displaystyle \mathbb {C} ^{n}} 1292:{\displaystyle \mathbb {C} ^{n}} 1263:{\displaystyle \mathbb {C} ^{n}} 954:{\displaystyle \mathbb {C} ^{n}} 873:{\displaystyle \mathbb {C} ^{n}} 840:{\displaystyle \mathbb {C} ^{n}} 811:{\displaystyle \mathbb {C} ^{n}} 750:{\displaystyle \mathbb {C} ^{n}} 556:{\displaystyle \mathbb {C} ^{n}} 299:than complex analytic geometry. 284:{\displaystyle \mathbb {C} ^{n}} 54:{\displaystyle \mathbb {C} ^{n}} 31199:10.1090/S0002-9904-1956-10018-9 31172:10.1090/S0002-9904-1956-10015-3 31145:10.1090/S0002-9904-1956-10013-X 30467:(3rd ed.), North Holland, 30265:Fuks, Boris Abramovich (1963). 29992:American Journal of Mathematics 29561:American Journal of Mathematics 29247:10.1090/S0002-9904-1967-11839-1 29129:American Journal of Mathematics 28213: 28138: 28105: 28087: 28060: 28033: 27947: 27912: 27865: 27765: 27722: 27614: 27587: 27545: 27511: 27467: 27443: 27397: 27364: 27321: 27282: 27197: 27173: 27061: 27028: 27001: 26974: 26923: 26890: 26855: 26766: 26732: 26467: 26449: 26361: 26244: 26203: 26162: 26133: 26100: 26049: 26006: 25911: 25876: 25835: 25800: 25761: 25720: 25657: 25633: 25600: 25379:10.1090/S0002-9904-1978-14483-8 24963: 24954: 24942: 24933: 24923: 24914: 24905: 24863: 24846: 24837: 24828: 24815: 24805: 24765: 24752: 24743: 24726: 24690:Infinite-dimensional holomorphy 24249:that induces an orientation on 24238:{\displaystyle X_{c}=f^{-1}(c)} 23226:{\displaystyle p\in \partial D} 23108:If the relative compact subset 23052:is a Stein manifold. i.e. Then 22107: 18851: 17970:has a neighborhood U such that 17859:{\displaystyle x\in \partial D} 17752: 17029:This is because once we have a 16541:{\displaystyle p\in \partial D} 15883:is positive-definite and class 13599:is also a domain of holomorphy. 13336:{\displaystyle V\not \subset D} 13155:is holomorphic, and the domain 12098:Weierstrass preparation theorem 11624: 11041:{\displaystyle \lambda (D^{*})} 9222:Weierstrass preparation theorem 8243:with piecewise smooth boundary 2301: 2184: 2178: 2120: 1900: 1894: 1078:may be represented by the real 459:will be harder to handle; when 405:, for every analytic function 333:Weierstrass preparation theorem 31255:Demailly, Jean-Pierre (2012). 29848:Hirzebruch, Friedrich (1966). 27008:Krantz, Steven George (2001). 25474:The Mathematical Intelligencer 25265: 25217: 24342: 24327: 24291: 24276: 24232: 24226: 24152: 24146: 24052: 24046: 23996: 23973:(which can be assumed to be a 23943:strongly pseudoconvex manifold 23763:Every domain of holomorphy in 23706: 23682: 23676: 23663: 23579: 23566: 23324: 23249: 23243: 23233:there is an open neighborhood 23060:is locally the Stein manifold. 22973: 22959: 22923: 22900: 22884: 22861: 22824: 22801: 22788: 22785: 22771: 22758: 22755: 22727: 22714: 22711: 22688: 22645: 22617: 22584:Another result, attributed to 22540: 22534: 22507:{\displaystyle f(x)\neq f(y).} 22498: 22492: 22483: 22477: 22454: 22448: 22357: 22351: 22325: 22321: 22315: 22308: 22284: 22280: 22274: 22267: 22240: 22163: 22157: 21793: 21779: 21717: 21703: 21663: 21649: 21636: 21633: 21619: 21597: 21594: 21573: 21560: 21557: 21543: 21510: 21496: 21425: 21395: 21378: 21352: 21338: 21279: 21258: 21220: 21199: 21166: 21145: 21132: 21129: 21091: 21066: 21045: 21006: 20968: 20943: 20922: 20654:) vanishes. In particular, by 20626: 20612: 20599: 20596: 20572: 20547: 20533: 20511:long exact cohomology sequence 20490: 20466: 20441: 20427: 20018: 19995: 19823:{\displaystyle {\mathcal {O}}} 19790: 19780: 19761: 19751: 19732: 19722: 19703: 19576: 19564: 19554: 19422: 19410: 19400: 19264:{\displaystyle {\mathcal {F}}} 19237:{\displaystyle {\mathcal {F}}} 19207: 19184: 19111: 19101: 19089: 19079: 19052: 19042: 18923:{\displaystyle {\mathcal {F}}} 18893: 18870: 18814: 18802: 18776: 18770: 18764: 18730: 18710: 18704: 18698: 18676: 18670: 18658: 18635: 18629: 18623: 18597: 18577: 18560: 18540: 18534: 18528: 18516: 18452: 18440: 18420: 18414: 18367: 18306:{\displaystyle 5\Rightarrow 1} 18297: 18287:are standard results. Proving 18280:{\displaystyle 4\Rightarrow 5} 18271: 18254:{\displaystyle 1\Rightarrow 4} 18245: 18219: 18213: 17996: 17990: 17928: 17773: 17767: 17726: 17720: 17697: 17685: 17673: 17667: 17634: 17626: 17619: 17607: 17575: 17569: 17532: 17524: 17517: 17505: 17473: 17467: 17393:is fixed in , and assume that 17369: 17361: 17315: 17307: 17280: 17268: 17140:{\displaystyle {\mathcal {B}}} 16923:{\displaystyle D_{k}\subset D} 16822: 16790: 16755: 16749: 16700: 16694: 16635: 16629: 16590: 16584: 16519:is pseudoconvex iff for every 16387: 16277: 16271: 16251: 16245: 16190: 16184: 16164: 16158: 16119: 16113: 16093: 16087: 16032: 16026: 15848: 15759: 15695: 15679: 15487: 15481: 15398: 15357: 15300: 15281:upper semi-continuous function 15176: 15161: 15155: 14942: 14886: 14873: 14839: 14826: 14816: 14801: 14652: 14631: 14563: 14544: 14511: 14486: 14476: 14465: 14446: 14354:{\displaystyle {\hat {K}}_{G}} 14339: 14246: 14182: 14176: 14131: 14125: 14103: 14099: 14093: 14086: 14062: 14058: 14052: 14045: 14012: 13944: 13938: 13199: 13187: 13019:idéal de domaines indéterminés 12885: 12872: 12829: 12810: 12805: 12799: 12778: 12736: 12624: 12603: 12578: 12569: 12555: 12546: 12521: 12509: 12468: 12459: 12445: 12436: 12411: 12399: 12364: 12356: 12339: 12331: 12306: 12294: 11981: 11963: 11953: 11938: 11906: 11891: 11871: 11845: 11756: 11741: 11727: 11712: 11520: 11493: 11474: 11447: 11276: 11272: 11257: 11237: 11222: 11212: 11206: 11200: 11194: 11112: 11080: 11035: 11022: 10841: 10809: 10788:it also contains the polydisc 10645: 10613: 10592:, the domain contains the set 10538: 10507: 10479: 10422: 10398: 10245: 10213: 10173: 10131: 10083: 10051: 9957:has an intersection part with 9890: 9884: 9851: 9845: 9726: 9708: 9579: 9573: 9546: 9540: 9507: 9501: 9432:in 1907 by showing that their 9334: 9292: 9121: 9095: 9066: 9040: 8954: 8942: 8936: 8930: 8908: 8902: 8864: 8852: 8843: 8837: 8808: 8796: 8790: 8784: 8762: 8756: 8627: 8587: 8549: 8534: 8515: 8502: 8494: 8482: 8473: 8461: 8435: 8417: 8373: 8361: 8351:the Bochner–Martinelli kernel 8075: 8067: 8023: 7985: 7805: 7799: 7771: 7756: 7584: 7578: 7550: 7535: 7517: 7504: 7452: 7446: 7368: 7360: 7325: 7280: 7246: 7240: 7157: 7129: 7107: 7062: 6983: 6955: 6933: 6888: 6841: 6814: 6795: 6768: 6662: 6638: 6179: 6152: 6133: 6106: 6017: 6011: 5981: 5975: 5931: 5899: 5807: 5780: 5755: 5728: 5723: 5691: 5630: 5617: 5551: 5524: 5505: 5478: 5389: 5383: 5306: 5278: 5256: 5211: 4998: 4953: 4875: 4867: 4798: 4770: 4748: 4703: 4560: 4533: 4508: 4481: 4476: 4444: 4383: 4370: 4267: 4222: 4095: 4069: 4063: 4037: 4032: 4000: 3939: 3926: 3914: 3882: 3795: 3769: 3763: 3737: 3734: 3708: 3703: 3658: 3549: 3536: 3500: 3474: 3471: 3445: 3440: 3382: 3310: 3297: 3236: 3191: 3137: 3105: 2932: 2887: 2870: 2858: 2588:{\displaystyle \gamma _{\nu }} 2455:{\displaystyle \gamma _{\nu }} 2372: 2002: 1962: 1956: 1927: 1921: 1891: 1885: 1859: 1853: 1716: 1647: 1565: 1559: 1517: 1505: 1496: 1490: 1481: 1475: 1466: 1454: 1424: 1195: 1186: 601:partial differential equations 430: 16:Type of mathematical functions 1: 31241:open source book by Jiří Lebl 30487:. American Mathematical Soc. 30107:Kyushu Journal of Mathematics 29916:"Le théorème de Riemann–Roch" 27232:Annales de l'Institut Fourier 27014:. American Mathematical Soc. 26062:Journal of Geometric Analysis 25555:Annales de l'Institut Fourier 25180: 24700: 23422: 21867:-dimensional manifold can be 19866:(Oka–Cartan) coherent theorem 18002:{\displaystyle \psi ^{-1}(U)} 17785:{\displaystyle Q(0)\subset D} 17738:{\displaystyle B(0)\subset D} 15879:When the hermitian matrix of 15033:, and for every complex line 14899:is a subharmonic function on 11296:in the real coordinate space 9953:, it would be that the above 9739:{\displaystyle f|_{W}=g|_{W}} 9436:have different dimensions as 3088:of one variable repeatedly, 1762:if and only if its real part 1307: 694:), for which it happens that 575:and their nature was to make 455:. Naturally the analogues of 31322:biholomorphically equivalent 31288:. License: Creative Commons 30231:Lectures on Riemann surfaces 30172: 30066:Kyoto Journal of Mathematics 28226:Oeuvres - Collected Papers I 27335:Annales Polonici Mathematici 27221:{\displaystyle \mathbb {C} } 25704:10.1017/CBO9781107325562.005 25166:{\displaystyle \mathbb {C} } 24734:zeroes of analytic functions 24436:{\displaystyle \mathbb {C} } 24110:invites a generalization of 23871:has the homotopy type of an 23013:relative compact open subset 21848:{\displaystyle \mathbb {C} } 21827:second axiom of countability 20166:, in terms of conditions on 19928:of holomorphic functions on 19484:{\displaystyle U\subseteq X} 18583:{\displaystyle (a,\delta ')} 17937:{\displaystyle \psi :X\to Y} 17866:there exist a neighbourhood 17171:in some neighborhood around 17167:which lies entirely outside 17096:If for every boundary point 16317: 15557: 14696:{\displaystyle \mathbb {C} } 14226:if for every compact subset 13073:is holomorpic on the domain 10969:A complete Reinhardt domain 9301:{\displaystyle \phi :U\to V} 8829: 8707: 8640: 8616: 8596: 5195:is holomorphic, on polydisc 4685:is holomorphic, on polydisc 2853: 2779: 2752: 2716: 2648: 2595:. Cartesian product closure 2328: 1366:{\displaystyle \mathbb {C} } 921:{\displaystyle \mathbb {R} } 899:, which gives its dimension 782:{\displaystyle \mathbb {C} } 721:The complex coordinate space 516:{\displaystyle \mathbb {C} } 228:complex projective varieties 95:, the functions studied are 81:has as a top-level heading. 31:the complex coordinate space 7: 31153:Chern, Shiing-Shen (1956). 30973:Encyclopedia of Mathematics 30955:Encyclopedia of Mathematics 30937:Encyclopedia of Mathematics 30919:Encyclopedia of Mathematics 30901:Encyclopedia of Mathematics 30883:Encyclopedia of Mathematics 30865:Encyclopedia of Mathematics 30847:Encyclopedia of Mathematics 30829:Encyclopedia of Mathematics 30824:"Plurisubharmonic function" 30822:Solomentsev, E.D. (2001) , 30811:Encyclopedia of Mathematics 30793:Encyclopedia of Mathematics 30775:Encyclopedia of Mathematics 30757:Encyclopedia of Mathematics 30750:Solomentsev, E.D. (2001) , 30739:Encyclopedia of Mathematics 30732:Solomentsev, E.D. (2001) , 30721:Encyclopedia of Mathematics 30714:Solomentsev, E.D. (2001) , 30703:Encyclopedia of Mathematics 30689:Encyclopedia of Mathematics 29470:10.1007/978-3-642-60925-1_1 29311:Encyclopedia of Mathematics 28722:Barth, Theodore J. (1968). 28521:Comptes Rendus Mathematique 28220:Serre, Jean-Pierre (2003). 28122:10.1007/978-3-642-69582-7_8 27931:10.1007/978-3-662-09873-8_2 27621:Fornæss, John Erik (1978). 27457:Encyclopedia of Mathematics 27187:Encyclopedia of Mathematics 27180:Solomentsev, E.D. (2001) , 26773:Serre, Jean-Pierre (1953). 26294:Tohoku Mathematical Journal 25895:10.1007/978-1-4757-1918-5_2 25848:Nagoya Mathematical Journal 25727:Poincare, M. Henri (1907). 25647:Encyclopedia of Mathematics 25640:Solomentsev, E.D. (2001) , 25517:10.1007/978-3-642-20554-5_5 24760:§Bochner–Martinelli formula 24653: 24494:Riemann's existence theorem 23731:The standard complex space 23178:is a singular Stein space, 23154:is itself a Stein manifold. 22223:holomorphically convex hull 21871:as a smooth submanifold of 21863:tells us that every smooth 20705:{\displaystyle f_{i}/f_{j}} 20273:{\displaystyle f_{i}-f_{j}} 18820:{\displaystyle (f,\delta )} 18458:{\displaystyle (f,\delta )} 15963:be a domain. One says that 13989:holomorphically convex hull 13841:Holomorphically convex hull 12232:{\displaystyle \Delta ^{2}} 12151:{\displaystyle \Delta ^{2}} 10755:if together with all point 9750:is said to be connected to 9247:-dimensional complex space 6678:such that the power series 6219:If a sequence of functions 4890:{\displaystyle |f|\leq {M}} 4634: 4164: 4158:Cauchy's evaluation formula 4147: 928:. Hence, as a set and as a 880:is also considered to be a 818:is a domain of holomorphy, 696:automorphic representations 105:so that, locally, they are 10: 31372: 31207:Remmert, Reinhold (1998). 30896:"Coherent algebraic sheaf" 30552:Krantz, Steven G. (1992). 30360:Forstnerič, Franc (2011). 30101:Matsumoto, Kazuko (2018). 29629:Kodaira, Kunihiko (1952). 29197:Remmert, Reinhold (1956). 28543:10.1016/j.crma.2010.11.020 28350:(3rd ed.), New York: 27708:10.4310/ICCM.2019.V7.N2.A2 26930:Range, R. Michael (2012). 26903:Kodai Mathematical Journal 26825:Forstnerič, Franc (2011). 26210:Sunada, Toshikazu (1978). 26126:10.1016/j.jmaa.2013.01.049 23585:{\displaystyle p=p(x_{0})} 23464:{\displaystyle x_{0}\in X} 23127:{\displaystyle D\subset M} 23033:{\displaystyle D\subset X} 22666:exponential sheaf sequence 22214:{\displaystyle K\subset X} 19306:, that is, every point in 18390: 18383:-problem(equation) with a 18173:is holomorphically convex. 18167:is a domain of holomorphy. 18117:is not weakly 1-complete. 15784:is positive semidefinite. 14786:{\displaystyle z_{1}\in D} 14707:be a positive function on 13980:{\displaystyle K\subset G} 13803:is a domain of holomorphy. 13275:{\displaystyle U\subset D} 13128:{\displaystyle \partial D} 10977:with regard to its centre 10781:{\displaystyle z^{0}\in D} 10585:{\displaystyle z^{0}\in D} 9919:{\displaystyle \partial U} 9786:{\displaystyle \partial U} 8259:{\displaystyle \partial D} 8201:Bochner–Martinelli formula 7252:{\displaystyle \omega (z)} 3848:{\displaystyle \partial D} 2357:complex differential forms 882:complex projective variety 599:of several variables, and 31351:Several complex variables 31280:Victor Guillemin. 18.117 31010:10.1007/978-1-4757-3849-0 30948:Onishchik, A.L. (2001) , 30878:"Coherent analytic sheaf" 30876:Onishchik, A.L. (2001) , 30858:Onishchik, A.L. (2001) , 30804:Onishchik, A.L. (2001) , 30641:10.1007/978-981-10-0291-5 30631:Noguchi, Junjiro (2016). 30588:10.1007/978-1-4757-1918-5 30370:10.1007/978-3-642-22250-4 30343:10.1007/978-3-642-20554-5 30209:Several Complex Variables 30190:10.1007/978-3-642-99659-7 29858:10.1007/978-3-642-62018-8 29827:10.1515/crll.1938.179.129 29512:10.1007/978-1-4757-3849-0 29304:Onishchik, A.L. (2001) , 29063:10.1007/978-4-431-55747-0 29018:Publicacions Matemàtiques 28832:Mathematische Zeitschrift 28809:10.1007/s00208-017-1539-x 28615:J. Math. Sci. Univ. Tokyo 28606:Noguchi, Junjiro (2011). 28488:10.1007/978-3-030-40120-7 28114:Coherent Analytic Sheaves 28070:Coherent Analytic Sheaves 28043:Coherent Analytic Sheaves 27989:Noguchi, Junjiro (2019), 27966:10.1007/978-4-431-55747-0 27784:10.1007/978-981-10-0291-5 27772:Noguchi, Junjiro (2016). 27741:10.1007/978-981-10-0291-5 27729:Noguchi, Junjiro (2016). 27497:21.11116/0000-0004-3A47-C 27450:Onishchik, A.L. (2001) , 26839:10.1007/978-3-642-22250-4 26750:10.1007/978-3-662-02661-8 26710:10.1007/s00407-009-0052-3 26435:10.1365/s13291-013-0061-7 25932:10.1080/17476930701747716 25861:10.1017/S0027763000013465 25784:10.1007/978-1-4684-7950-8 25772:. In Wu, Hung-Hsi (ed.). 25321:10.1007/s13373-014-0058-2 25302:Błocki, Zbigniew (2014). 24821:When described using the 24591:of enough high-dimension 24574:Kodaira embedding theorem 24564:(was first introduced by 24502:coherent sheaf cohomology 24498:Kodaira embedding theorem 24376:Cartan's theorems A and B 23799:is a Stein manifold, too. 22594:holomorphic vector bundle 22396:holomorphically separable 22104:of an analytic function. 21861:Whitney embedding theorem 20836:, and the quotient sheaf 20135:Cartan's theorems A and B 19957:, or the structure sheaf 19517:, and arbitrary morphism 18985:has an open neighborhood 18841:coherent sheaf cohomology 16223:if there exists a smooth 15976:plurisubharmonic function 14681:be a Hartogs's domain on 14665:is domain of holomorphy. 14524:for every compact subset 13922:complex analytic manifold 13022:is interpreted theory of 12270:Thullen's classic results 10987:Cauchy's integral theorem 3086:Cauchy's integral formula 961:may be identified to the 893:-dimensional vector space 707:edge-of-the-wedge theorem 681:totally real number field 71:several complex variables 30134:Takeuchi, Akira (1964). 30079:10.1215/21562261-1625181 29701:"Un théorème de dualité" 29347:10.2977/prims/1195183303 29030:10.5565/PUBLMAT_45201_11 28073:. Springer. p. 84. 28046:. Springer. p. 60. 27390:10.2977/prims/1195186709 27035:Lempert, Laszlo (1981). 25626:10.2977/prims/1195181825 25429:10.11650/twjm/1500407292 25202:Hörmander, Lars (1965). 24795:{\displaystyle D_{\nu }} 24589:complex projective space 24504:, and also Serre proved 24021:, such that the subsets 23875:-dimensional CW-Complex. 23333:{\displaystyle f:M\to N} 20147:Mittag-Leffler's theorem 19693:If in an exact sequence 19640:{\displaystyle \varphi } 19627:-modules, the kernel of 19442:for some natural number 18199:is Locally pseudoconvex. 17657:is called its shell. If 17382:{\displaystyle |u|<1} 17042:{\displaystyle \varphi } 15990:{\displaystyle \varphi } 14160:polynomially convex hull 13450:case, the every domain ( 9308:and the inverse mapping 9218:inverse function theorem 9176:defined by the equation 8286:denotes the exterior or 7226:Laurent series expansion 2737:. Also, take the closed 2561:{\displaystyle D_{\nu }} 1804:Cauchy-Riemann equations 1694:Cauchy–Riemann equations 1227:square of absolute value 727:complex coordinate space 651:in it, meaning that the 335:would now be classed as 321:Jacobi inversion problem 134:Cauchy–Riemann equations 31249:OpenContent book See B2 31180:Zariski, Oscar (1956). 30966:Parshin, A.N. (2001) , 30894:Danilov, V.I. (2001) , 30840:Danilov, V.I. (2001) , 30734:"Biholomorphic mapping" 30672:. Courier Corporation. 30120:10.2206/kyushujm.72.107 28301:10.32917/hmj/1558576819 28268:10.32917/hmj/1558749869 28206:10.32917/hmj/1558490525 28145:Cousin, Pierre (1895). 28094:Demailly, Jean-Pierre. 27295:Manuscripta Mathematica 27166:10.2206/kyushumfs.13.37 27147:Kajiwara, Joji (1959). 27087:10.2206/kyushumfs.41.45 27068:Shon, Kwang Ho (1987). 26347:10.4099/jjm1924.23.0_97 26169:Thullen, Peter (1931). 23308:an injective, and also 23281:{\displaystyle U\cap D} 23204:. Suppose that for all 22417:{\displaystyle x\neq y} 20764:{\displaystyle f/f_{i}} 20368:; in other words, that 20330:{\displaystyle f-f_{i}} 18498:{\displaystyle \delta } 18110:{\displaystyle \Omega } 18090:{\displaystyle \Omega } 17897:{\displaystyle U\cap D} 17450:{\displaystyle \Delta } 17092:Levi total pseudoconvex 16219:-dimensional manifold. 15462:denotes the unit disk. 14724:{\displaystyle \Omega } 14419:{\displaystyle n\geq 1} 13829:{\displaystyle n\geq 3} 13055:{\displaystyle \Omega } 12855:{\displaystyle \sigma } 12065:is a compact subset of 10031:{\displaystyle n\geq 1} 9946:{\displaystyle n\geq 2} 9689:{\displaystyle U\cap V} 9202:{\displaystyle z_{1}=0} 8441:{\displaystyle (n,n-1)} 6870:converges uniformly at 3863:can be calculated as a 2428:{\displaystyle \gamma } 2257:Using the formalism of 2040:and the imaginary part 1782:and its imaginary part 619:. The celebrated paper 31356:Multivariable calculus 31310:Holomorphically convex 31126:Martin, W. T. (1956). 30930:Chirka, E.M. (2001) , 30912:Chirka, E.M. (2001) , 30842:"Quasi-coherent sheaf" 30788:"Domain of holomorphy" 30768:Chirka, E.M. (2001) , 30635:. p. XVIII, 397. 30389:Theory of Stein spaces 30305:Cartan, Henri (1992). 30060:Ohsawa, Takeo (2012). 29396:Miranda, Rick (1995). 29328:Ohsawa, Takeo (1982). 29228:Forster, Otto (1967). 28728:Trans. Amer. Math. Soc 28689:Cartan, Henri (1957). 28650:Grauert, Hans (1955). 28321:Séminaire Henri Cartan 28147:"Sur les fonctions de 28020:10.2996/kmj/1572487232 27778:. p. XVIII, 397. 27735:. p. XVIII, 397. 27646: 27371:Ohsawa, Takeo (1981). 27328:Ohsawa, Takeo (2012). 27222: 27204:Ohsawa, Takeo (2018). 26916:10.2996/kmj/1138845123 26601:Cartan, Henri (1950). 25842:Sakai, Eiichi (1970). 25768:Siu, Yum-Tong (1991). 25607:Ohsawa, Takeo (1984). 25410:Chen, So-Chin (2000). 25360:Siu, Yum-Tong (1978). 25272:Ohsawa, Takeo (2002). 25227: 25167: 25145: 25080: 25051: 24892: 24796: 24644: 24554: 24486: 24437: 24349: 24301: 24239: 24168: 24104: 24078:for every real number 24068: 24015: 23963: 23917: 23916:{\displaystyle x\in X} 23842: 23786: 23754: 23716: 23613: 23586: 23544: 23511: 23465: 23415:is itself Stein ? 23405: 23348:is itself Stein ? 23334: 23282: 23256: 23227: 23198: 23160:complex analytic space 23128: 23087: 23034: 22986: 22936: 22831: 22658: 22547: 22508: 22461: 22418: 22372: 22215: 22190: 22170: 22137: 22089: 22029: 21992: 21963: 21926: 21897: 21849: 21806: 21756: 21755:{\displaystyle q>0} 21730: 21673: 21517: 21469: 21435: 21359: 21315: 21286: 21233: 21173: 21016: 20896: 20876: 20826: 20797: 20765: 20726: 20706: 20633: 20500: 20382: 20362: 20331: 20294: 20274: 20234: 20207: 20127: 20089: 20060: 20025: 19982: 19951: 19918: 19872: 19855: 19824: 19800: 19676: 19641: 19621: 19590: 19511: 19510:{\displaystyle n>0} 19485: 19456: 19436: 19363: 19343: 19320: 19300: 19265: 19246: 19238: 19214: 19164: 19144: 19121: 18999: 18979: 18955: 18924: 18900: 18821: 18786: 18717: 18642: 18584: 18547: 18507: 18499: 18479: 18459: 18427: 18377: 18344: 18307: 18281: 18255: 18229: 18162: 18154: 18111: 18091: 18071: 18034: 18003: 17964: 17963:{\displaystyle y\in Y} 17938: 17898: 17860: 17823: 17786: 17739: 17704: 17651: 17549: 17451: 17431: 17383: 17347: 17287: 17205: 17185: 17161: 17141: 17110: 17074: 17043: 17020: 17002: 16955: 16924: 16876: 16838: 16732: 16676: 16619: 16562: 16542: 16509: 16468: 16430: 16399: 16352: 16307: 16306:{\displaystyle H\psi } 16284: 16197: 16126: 16051: 15991: 15957: 15908: 15861: 15812: 15778: 15650: 15630: 15599: 15579: 15494: 15493:{\displaystyle u=u(z)} 15456: 15434:is subharmonic, where 15425: 15367: 15327: 15273: 15245: 15183: 15135: 15094: 15016: 14974: 14905: 14893: 14846: 14787: 14754: 14725: 14697: 14659: 14618: 14570: 14518: 14420: 14391: 14355: 14319: 14299: 14262: 14224:holomorphically convex 14216: 14189: 14149: 13981: 13951: 13914: 13894: 13874: 13830: 13797: 13757: 13730: 13696: 13681: 13642: 13593: 13578: 13539: 13472: 13444: 13411: 13369: 13337: 13311: 13276: 13246: 13206: 13129: 13102: 13066: 13056: 13006: 12977: 12944: 12905: 12856: 12836: 12755: 12705: 12678: 12631: 12491: 12381: 12260: 12233: 12206: 12179: 12152: 12125: 12048: 11988: 11805: 11773: 11657: 11615: 11589: 11543: 11410: 11351: 11319: 11283: 11166: 11042: 11010:logarithmically convex 10997:Logarithmically-convex 10960: 10782: 10738: 10586: 10550: 10429: 10364: 10186: 10105: 10032: 10006: 9947: 9920: 9897: 9858: 9815: 9793:: there exists domain 9787: 9740: 9690: 9660: 9640: 9586: 9553: 9514: 9475: 9414: 9413:{\displaystyle n>1} 9364: 9344: 9302: 9270: 9203: 9170: 9141: 9079: 9012: 8965: 8875: 8738: 8442: 8404: 8403:{\displaystyle \zeta } 8380: 8345: 8312: 8311:{\displaystyle \zeta } 8280: 8279:{\displaystyle \land } 8260: 8237: 8185: 8142: 8099: 8034: 7716: 7482: 7423: 7259:be holomorphic in the 7253: 7212: 7038: 6864: 6731: 6672: 6601: 6481: 6325: 6294: 6259: 6205: 6069: 5988: 5959: 5870: 5441: 5359: 5170: 4891: 4853: 4672: 4622: 4135: 3849: 3823: 3078: 3034: 2835: 2761: 2731: 2657: 2589: 2562: 2535: 2497:Jordan closed curve. ( 2491: 2456: 2429: 2391: 2349: 2249: 2074: 2054: 2034: 2014: 1969: 1820: 1802:satisfy the so-called 1796: 1776: 1756: 1728: 1654: 1524: 1436: 1393: 1392:{\displaystyle z\in D} 1367: 1345: 1293: 1264: 1212: 1137: 990: 955: 922: 874: 841: 812: 783: 751: 557: 517: 442: 303:Historical perspective 285: 256: 216: 162: 55: 31216:Séminaires et Congrès 30968:"Finiteness theorems" 30614:10.1007/3-7643-7491-8 30508:. Walter de Gruyter. 30153:10.2969/jmsj/01620159 30027:Annals of Mathematics 29957:Annals of Mathematics 29656:10.1073/pnas.38.6.522 29596:Annals of Mathematics 29461:Algebraic Geometry II 29363:Annals of Mathematics 29261:Simha, R. R. (1989). 29164:Annals of Mathematics 28951:Mathematische Annalen 28916:Mathematische Annalen 28874:Annals of Mathematics 28787:Mathematische Annalen 28656:Mathematische Annalen 28431:Mathematische Annalen 28389:Mathematische Annalen 28282:Oka, Kiyoshi (1937). 28249:Oka, Kiyoshi (1936). 28187:Oka, Kiyoshi (1939). 27832:Annals of Mathematics 27656:Mathematische Annalen 27647: 27520:Annals of Mathematics 27476:Mathematische Annalen 27410:Mathematische Annalen 27223: 27123:10.2969/jmsj/00620177 26864:Mathematische Annalen 26581:10.2969/jmsj/00320259 26560:10.2969/jmsj/00310204 26520:Oka, Kiyoshi (1961). 26494:Oka, Kiyoshi (1950). 26370:Mathematische Annalen 26287:Oka, Kiyoshi (1943), 26257:Mathematische Annalen 26216:Mathematische Annalen 26175:Mathematische Annalen 25775:Contemporary Geometry 25642:"Weierstrass theorem" 25228: 25168: 25146: 25081: 25052: 24893: 24797: 24645: 24555: 24487: 24438: 24372:holomorphic functions 24350: 24302: 24240: 24169: 24105: 24103:{\displaystyle \psi } 24090:(1911). The function 24069: 24016: 23964: 23962:{\displaystyle \psi } 23918: 23893:Every Stein manifold 23843: 23810:can be embedded into 23806:of complex dimension 23787: 23755: 23717: 23614: 23612:{\displaystyle x_{0}} 23587: 23545: 23512: 23466: 23434:is complex manifold, 23406: 23360:be a Stein space and 23335: 23304:be a Stein space and 23283: 23257: 23228: 23199: 23140:strongly pseudoconvex 23129: 23088: 23035: 22987: 22937: 22832: 22659: 22581:is a Stein manifold. 22548: 22509: 22462: 22419: 22373: 22216: 22171: 22138: 22123:of complex dimension 22102:analytic continuation 22090: 22030: 21993: 21964: 21927: 21898: 21850: 21807: 21757: 21731: 21674: 21518: 21470: 21436: 21360: 21316: 21287: 21241:The cohomology group 21234: 21174: 21017: 20897: 20895:{\displaystyle \phi } 20877: 20827: 20798: 20766: 20727: 20707: 20666:Second Cousin problem 20662:is a Stein manifold. 20634: 20501: 20383: 20363: 20361:{\displaystyle U_{i}} 20332: 20295: 20275: 20235: 20233:{\displaystyle U_{i}} 20208: 20206:{\displaystyle f_{i}} 20128: 20090: 20061: 20026: 19983: 19952: 19919: 19856: 19825: 19801: 19677: 19642: 19622: 19591: 19512: 19486: 19457: 19437: 19364: 19344: 19321: 19301: 19266: 19239: 19215: 19165: 19145: 19122: 19005:in which there is an 19000: 18980: 18956: 18925: 18901: 18822: 18787: 18718: 18643: 18585: 18548: 18500: 18480: 18460: 18428: 18378: 18345: 18308: 18282: 18256: 18230: 18155: 18112: 18092: 18072: 18035: 18004: 17965: 17939: 17899: 17861: 17824: 17787: 17740: 17705: 17652: 17550: 17452: 17432: 17384: 17348: 17288: 17206: 17204:{\displaystyle \rho } 17186: 17184:{\displaystyle \rho } 17162: 17160:{\displaystyle \rho } 17142: 17111: 17109:{\displaystyle \rho } 17080:exhaustion function. 17075: 17044: 17021: 16982: 16956: 16925: 16877: 16839: 16706: 16677: 16599: 16563: 16543: 16510: 16469: 16431: 16400: 16353: 16308: 16285: 16198: 16127: 16065:for all real numbers 16052: 15992: 15958: 15909: 15862: 15813: 15779: 15651: 15631: 15600: 15580: 15495: 15457: 15426: 15368: 15328: 15274: 15246: 15184: 15136: 15095: 15031:upper semi-continuous 15017: 14975: 14911:looks like a kind of 14894: 14847: 14788: 14755: 14726: 14698: 14675: 14660: 14619: 14571: 14519: 14421: 14392: 14356: 14320: 14300: 14263: 14217: 14190: 14150: 13982: 13952: 13915: 13895: 13875: 13831: 13798: 13758: 13756:{\displaystyle D_{2}} 13731: 13729:{\displaystyle D_{1}} 13697: 13661: 13643: 13594: 13558: 13540: 13473: 13445: 13412: 13370: 13338: 13312: 13277: 13247: 13207: 13130: 13103: 13057: 13041: 13007: 12978: 12945: 12906: 12857: 12837: 12756: 12706: 12704:{\displaystyle G_{2}} 12679: 12677:{\displaystyle G_{1}} 12632: 12492: 12382: 12261: 12234: 12207: 12180: 12153: 12126: 12049: 11989: 11806: 11774: 11658: 11616: 11590: 11544: 11364: 11352: 11328:Every such domain in 11320: 11284: 11167: 11043: 10961: 10783: 10739: 10587: 10551: 10430: 10365: 10187: 10106: 10033: 10007: 9948: 9921: 9898: 9859: 9816: 9788: 9741: 9691: 9661: 9641: 9587: 9554: 9515: 9476: 9444:Analytic continuation 9426:biholomorphic mapping 9415: 9365: 9363:{\displaystyle \phi } 9345: 9303: 9271: 9204: 9171: 9142: 9080: 9013: 8966: 8876: 8739: 8443: 8405: 8381: 8346: 8313: 8281: 8266:, and let the symbol 8261: 8238: 8186: 8143: 8100: 8035: 7696: 7462: 7424: 7254: 7213: 7039: 6865: 6685: 6673: 6602: 6461: 6326: 6324:{\displaystyle f_{v}} 6295: 6293:{\displaystyle f_{v}} 6269:, the limit function 6260: 6206: 6023: 5989: 5960: 5871: 5395: 5360: 5171: 4892: 4854: 4673: 4623: 4136: 3850: 3824: 3079: 3035: 2836: 2762: 2732: 2658: 2590: 2563: 2536: 2492: 2457: 2430: 2392: 2350: 2259:Wirtinger derivatives 2250: 2075: 2055: 2035: 2015: 1970: 1821: 1797: 1777: 1757: 1729: 1655: 1525: 1437: 1394: 1368: 1346: 1303:Holomorphic functions 1294: 1265: 1239:holomorphic functions 1213: 1138: 998:topological dimension 991: 963:real coordinate space 956: 923: 888:, etc. It is also an 875: 847:can be regarded as a 842: 813: 784: 752: 665:Hilbert modular forms 558: 518: 495:analytic continuation 443: 329:mathematical analysis 317:hypergeometric series 286: 257: 224:meromorphic functions 217: 163: 56: 31314:Domain of holomorphy 30413:Шабат, Б.В. (1985). 28472:. pp. 133–192. 28151:variables complexes" 27627: 27602:(249/262): 178–183. 27210: 26811:on October 20, 2020. 26741:Sheaves on Manifolds 25698:. pp. 134–186. 25208: 25155: 25126: 25061: 25017: 24873: 24823:domain of holomorphy 24779: 24675:Dolbeault cohomology 24622: 24576:says that a compact 24532: 24447: 24425: 24415:Riemann's inequality 24411:Riemann-Roch theorem 24311: 24260: 24197: 24122: 24094: 24025: 23981: 23953: 23901: 23882:is a Stein manifold 23814: 23792:is a Stein manifold. 23767: 23760:is a Stein manifold. 23735: 23623: 23596: 23554: 23525: 23475: 23442: 23364: 23312: 23266: 23255:{\displaystyle U(p)} 23237: 23208: 23182: 23112: 23068: 23018: 22946: 22848: 22675: 22604: 22524: 22471: 22432: 22428:, then there exists 22402: 22231: 22199: 22147: 22127: 22070: 22010: 21973: 21944: 21907: 21875: 21837: 21766: 21740: 21690: 21530: 21483: 21448: 21372: 21325: 21296: 21245: 21186: 21032: 20909: 20886: 20840: 20807: 20778: 20740: 20716: 20674: 20520: 20414: 20372: 20345: 20308: 20284: 20244: 20217: 20190: 20178:First Cousin problem 20106: 20070: 20037: 19992: 19961: 19932: 19878: 19834: 19810: 19697: 19655: 19631: 19600: 19521: 19495: 19469: 19446: 19373: 19353: 19333: 19310: 19279: 19251: 19224: 19181: 19154: 19134: 19015: 18989: 18969: 18934: 18910: 18867: 18858:quasi-coherent sheaf 18799: 18727: 18655: 18594: 18557: 18513: 18489: 18469: 18437: 18411: 18395:The introduction of 18358: 18325: 18291: 18265: 18239: 18207: 18181:analytic polyhedrons 18129: 18101: 18081: 18052: 18013: 17974: 17948: 17916: 17882: 17878:holomorphic. ( i.e. 17841: 17804: 17800:Riemann domain over 17761: 17714: 17661: 17563: 17461: 17441: 17397: 17357: 17297: 17236: 17224:Family of Oka's disk 17195: 17175: 17151: 17127: 17100: 17053: 17033: 16973: 16934: 16901: 16855: 16688: 16575: 16552: 16523: 16478: 16440: 16409: 16366: 16331: 16294: 16231: 16144: 16073: 16005: 15981: 15930: 15887: 15826: 15791: 15663: 15640: 15609: 15589: 15508: 15469: 15438: 15380: 15345: 15288: 15263: 15197: 15149: 15104: 15040: 14989: 14930: 14909:subharmonic function 14856: 14797: 14764: 14735: 14715: 14685: 14677:Hartogs (1906): Let 14673:Hartogs showed that 14628: 14580: 14536: 14438: 14404: 14369: 14329: 14309: 14283: 14230: 14206: 14166: 14002: 13965: 13961:. For a compact set 13928: 13904: 13884: 13849: 13814: 13767: 13740: 13713: 13704:Behnke–Stein theorem 13652: 13606: 13549: 13503: 13454: 13428: 13395: 13347: 13321: 13286: 13260: 13254:domain of holomorphy 13227: 13163: 13116: 13077: 13046: 13034:Domain of holomorphy 12987: 12983:, later extended to 12958: 12925: 12866: 12846: 12765: 12715: 12688: 12661: 12503: 12393: 12288: 12243: 12216: 12189: 12162: 12135: 12108: 12029: 12006:holomorphic function 11820: 11783: 11671: 11638: 11599: 11553: 11361: 11332: 11300: 11182: 11055: 11016: 10991:Jordan curve theorem 10795: 10759: 10599: 10563: 10443: 10384: 10199: 10115: 10042: 10016: 9981: 9931: 9907: 9868: 9829: 9805: 9774: 9700: 9674: 9650: 9600: 9563: 9524: 9485: 9456: 9398: 9354: 9312: 9280: 9251: 9180: 9151: 9089: 9034: 8987: 8895: 8749: 8455: 8414: 8394: 8355: 8326: 8302: 8270: 8247: 8218: 8152: 8109: 8047: 7436: 7266: 7234: 7048: 6874: 6682: 6619: 6347: 6308: 6277: 6223: 6005: 5987:{\displaystyle f(z)} 5969: 5890: 5372: 5199: 4904: 4863: 4689: 4651: 4177: 3876: 3836: 3095: 3044: 2845: 2771: 2744: 2667: 2599: 2572: 2545: 2501: 2470: 2439: 2419: 2363: 2265: 2084: 2064: 2044: 2024: 1981: 1830: 1810: 1786: 1766: 1738: 1702: 1553: 1448: 1403: 1377: 1355: 1320: 1316:defined on a domain 1274: 1245: 1156: 1088: 1064:by a complex number 968: 936: 910: 855: 822: 793: 771: 732: 711:quantum field theory 669:Siegel modular forms 635:géometrie algébrique 631:géometrie analytique 538: 505: 409: 399:isolated singularity 266: 234: 176: 170:domain of holomorphy 144: 88:, which is the case 36: 31334:exhaustion function 31286:https://ocw.mit.edu 30698:"Analytic function" 30248:Фукс, Б.А. (1962). 29647:1952PNAS...38..522K 28101:. Institut Fourier. 27182:"Riemannian domain" 25672:(98/103): 262–270. 23797:of a Stein manifold 23288:is Stein space. Is 22754: 22644: 21411: 21078: 20955: 20559: 20453: 20213:along with domains 19689:(1955) proves that 19551: 19397: 19076: 19039: 18590:is arbitrary, then 18426:{\displaystyle (I)} 17389:when the parameter 17191:, except the point 15869:positive (1,1)-form 14298:{\displaystyle n=1} 14268:is also compact in 14109: for all 13443:{\displaystyle n=1} 13410:{\displaystyle f=g} 13215:Formally, a domain 13062:in the figure with 11816:Internal domain of 11614:{\displaystyle a=0} 11004:A Reinhardt domain 10902: 10747:A Reinhardt domain 10524: 10306: 10172: 10148: 9668:connected component 9434:automorphism groups 8167: 8124: 8094: 8062: 7934: 7903: 7390: for all 7179: for all 7005: for all 5328: for all 5181:Liouville's theorem 4820: for all 3073: 3001: for all 2767:so that it becomes 1351:and with values in 1012:a complex structure 485:, and Germany with 337:commutative algebra 30998:Algebraic Geometry 30752:"Reinhardt domain" 29933:10.24033/bsmf.1500 29891:10.1007/BFb0066283 29786:10.1007/BFb0093697 29717:10.1007/BF02564268 29697:Serre, Jean-Pierre 29499:Algebraic Geometry 28963:10.1007/BF01470950 28928:10.1007/BF01451029 28844:10.1007/BF01111528 28793:(3–4): 1047–1067. 28708:10.24033/bsmf.1481 28668:10.1007/BF01362369 28578:10.1007/bf02054949 28443:10.1007/BF01360812 28401:10.1007/BF01447838 28352:Dover Publications 28168:10.1007/BF02402869 27890:10.1007/bf02684778 27821:Serre, Jean-Pierre 27668:10.1007/BF01420649 27642: 27566:10.1007/BF02564357 27488:10.1007/BF01343146 27422:10.1007/BF01343548 27349:10.4064/ap106-0-19 27307:10.1007/BF01312449 27218: 27054:10.24033/bsmf.1948 26876:10.1007/BF01597355 26620:10.24033/bsmf.1409 26513:10.24033/bsmf.1408 26383:10.1007/BF01360125 26270:10.1007/BF01455905 26229:10.1007/BF01405009 26188:10.1007/bf01457933 26085:10.1007/BF02922095 25746:10.1007/BF03013518 25546:Serre, Jean-Pierre 25509:Complex Analysis 2 25486:10.1007/BF03026112 25250:10.1007/BF02391775 25223: 25163: 25141: 25076: 25047: 24888: 24792: 24736:; this is not the 24685:Harmonic morphisms 24640: 24615:deformation theory 24595:. In addition the 24550: 24482: 24433: 24345: 24297: 24235: 24164: 24100: 24064: 24011: 23959: 23913: 23838: 23782: 23750: 23712: 23609: 23582: 23540: 23507: 23461: 23401: 23390: 23330: 23278: 23252: 23223: 23194: 23189:⊂ ⊂ 23124: 23083: 23030: 22982: 22932: 22842:Cartan's theorem B 22827: 22736: 22654: 22626: 22543: 22504: 22457: 22424:are two points in 22414: 22368: 22306: 22211: 22166: 22133: 22085: 22025: 21988: 21959: 21922: 21893: 21845: 21802: 21752: 21726: 21669: 21513: 21465: 21431: 21355: 21311: 21282: 21229: 21169: 21012: 20892: 20872: 20822: 20793: 20761: 20722: 20702: 20656:Cartan's theorem B 20629: 20496: 20378: 20358: 20327: 20290: 20270: 20230: 20203: 20123: 20085: 20056: 20021: 19978: 19947: 19914: 19851: 19820: 19796: 19672: 19647:is of finite type. 19637: 19617: 19586: 19530: 19507: 19481: 19465:for each open set 19452: 19432: 19376: 19359: 19339: 19316: 19296: 19261: 19234: 19210: 19177:on a ringed space 19160: 19140: 19117: 19055: 19018: 18995: 18975: 18951: 18920: 18896: 18817: 18782: 18713: 18638: 18580: 18543: 18495: 18475: 18455: 18423: 18373: 18340: 18303: 18277: 18251: 18225: 18150: 18107: 18087: 18067: 18030: 17999: 17960: 17934: 17894: 17856: 17819: 17782: 17735: 17700: 17647: 17545: 17447: 17427: 17379: 17343: 17283: 17201: 17181: 17157: 17137: 17120:, there exists an 17106: 17070: 17039: 17016: 16951: 16920: 16872: 16834: 16672: 16558: 16538: 16505: 16464: 16426: 16395: 16348: 16303: 16280: 16193: 16122: 16059:relatively compact 16047: 16001:such that the set 15987: 15971:if there exists a 15953: 15904: 15857: 15808: 15774: 15646: 15626: 15595: 15575: 15490: 15452: 15421: 15363: 15323: 15269: 15241: 15179: 15131: 15090: 15012: 14970: 14889: 14842: 14783: 14750: 14721: 14711:such that the set 14693: 14655: 14614: 14566: 14514: 14416: 14387: 14351: 14315: 14295: 14258: 14212: 14185: 14145: 14084: 13977: 13947: 13910: 13890: 13870: 13826: 13793: 13753: 13726: 13692: 13638: 13589: 13535: 13482:everywhere on the 13468: 13440: 13407: 13365: 13333: 13307: 13272: 13242: 13202: 13125: 13098: 13067: 13052: 13002: 12973: 12940: 12901: 12852: 12832: 12751: 12701: 12674: 12627: 12487: 12377: 12256: 12229: 12202: 12175: 12148: 12121: 12044: 11984: 11801: 11769: 11653: 11611: 11585: 11539: 11347: 11315: 11279: 11175:under the mapping 11162: 11038: 10989:without using the 10956: 10888: 10778: 10734: 10582: 10546: 10510: 10425: 10360: 10292: 10182: 10158: 10134: 10101: 10028: 10002: 9943: 9916: 9893: 9854: 9811: 9783: 9736: 9686: 9656: 9636: 9582: 9549: 9510: 9471: 9410: 9360: 9340: 9298: 9266: 9199: 9166: 9137: 9075: 9008: 8961: 8960: 8871: 8870: 8734: 8586: 8438: 8400: 8376: 8341: 8308: 8276: 8256: 8233: 8181: 8155: 8138: 8112: 8095: 8082: 8050: 8030: 8028: 7907: 7876: 7419: 7249: 7208: 7034: 6860: 6668: 6597: 6321: 6290: 6255: 6201: 5984: 5955: 5866: 5864: 5355: 5166: 4887: 4849: 4668: 4618: 4131: 3845: 3819: 3817: 3074: 3053: 3030: 2831: 2757: 2727: 2653: 2585: 2558: 2531: 2487: 2452: 2425: 2412:. Each disk has a 2387: 2345: 2245: 2070: 2050: 2030: 2010: 1965: 1816: 1792: 1772: 1752: 1724: 1650: 1520: 1432: 1389: 1363: 1341: 1289: 1260: 1208: 1133: 1124: 1014:is specified by a 986: 951: 918: 870: 837: 808: 779: 747: 675:(respectively the 586:algebraic geometry 553: 513: 438: 297:algebraic geometry 281: 252: 212: 158: 51: 31326:Levi pseudoconvex 31273:978-2-7302-1610-4 31119:978-3-662-43412-3 31019:978-0-387-90244-9 30994:Hartshorne, Robin 30932:"Cousin problems" 30770:"Hartogs theorem" 30650:978-981-10-0289-2 30597:978-1-4419-3078-1 30571:978-0-8218-2724-6 30494:978-0-8218-2165-7 30474:978-1-493-30273-4 30439:10.1090/mmono/110 30379:978-3-642-22249-8 30352:978-3-642-20554-5 30329:Freitag, Eberhard 30276:978-1-4704-4428-0 30218:978-0-598-34865-4 30199:978-3-642-98844-8 29959:. Second Series. 29900:978-3-540-05647-8 29867:978-3-540-58663-0 29809:Weil, A. (1938). 29795:978-3-540-61018-2 29521:978-0-387-90244-9 29494:Hartshorne, Robin 29479:978-3-642-64607-2 29166:. Second Series. 28876:, Second Series, 28497:978-3-030-40119-1 28361:978-0-486-47004-7 28235:978-3-642-39815-5 28131:978-3-642-69584-1 28080:978-3-642-69582-7 28053:978-3-642-69582-7 27940:978-3-642-08150-7 27793:978-981-10-0289-2 27750:978-981-10-0289-2 27522:, Second Series, 26848:978-3-642-22249-8 26827:"Stein Manifolds" 26759:978-3-642-08082-1 25904:978-1-4419-3078-1 25828:978-3-03719-049-4 25793:978-1-4684-7950-8 25526:978-3-642-20553-8 25285:978-1-4704-4636-9 25220: 25097:polyhedral domain 24738:analytic geometry 24562:vanishing theorem 24461: 24247:contact structure 23999: 23681: 23373: 23296:more generalized 22334: 22291: 22243: 22136:{\displaystyle n} 22121:complex manifolds 21412: 21079: 20956: 20725:{\displaystyle f} 20560: 20454: 20381:{\displaystyle f} 20293:{\displaystyle f} 19687:Jean-Pierre Serre 19455:{\displaystyle n} 19362:{\displaystyle X} 19342:{\displaystyle U} 19328:open neighborhood 19319:{\displaystyle X} 19163:{\displaystyle J} 19143:{\displaystyle I} 18998:{\displaystyle U} 18978:{\displaystyle X} 18478:{\displaystyle f} 18370: 18203:The implications 17684: 17425: 17353:, holomorphic in 17293:be continuous on 16825: 16797: 16793: 16654: 16561:{\displaystyle w} 16243: 16156: 16085: 15851: 15837: 15772: 15762: 15649:{\displaystyle u} 15598:{\displaystyle u} 15563: 15560: 15272:{\displaystyle X} 14542: 14489: 14474: 14444: 14342: 14318:{\displaystyle G} 14249: 14215:{\displaystyle G} 14110: 14069: 14015: 13913:{\displaystyle n} 13893:{\displaystyle G} 13186: 12637:(Thullen domain). 12590: 12544: 12434: 12354: 12329: 11962: 11924: 11918: 11538: 10926: 10707: 10330: 9814:{\displaystyle f} 9659:{\displaystyle W} 9620: 9611: 9214:maximal principle 8884:In particular if 8832: 8710: 8643: 8619: 8599: 8565: 8563: 8525: 8388:differential form 8294:is in the domain 7984: 7980: 7871: 7870: 7748: 7730: 7641: 7623: 7527: 7496: 7391: 7180: 7006: 6859: 6637: 6595: 6456: 6197: 5830: 5640: 5569: 5329: 5164: 5065: 4821: 4642: 4641: 4583: 4393: 4334: 4155: 4154: 4099: 3949: 3865:multiple integral 3861:iterated integral 3799: 3559: 3504: 3320: 3265: 3163: 3002: 2856: 2782: 2755: 2719: 2651: 2414:rectifiable curve 2375: 2334: 2331: 2243: 2210: 2182: 2176: 2146: 2073:{\displaystyle f} 2053:{\displaystyle v} 2033:{\displaystyle u} 1954: 1919: 1898: 1883: 1851: 1819:{\displaystyle p} 1795:{\displaystyle v} 1775:{\displaystyle u} 930:topological space 759:Cartesian product 609:complex manifolds 597:automorphic forms 593:analytic geometry 457:contour integrals 391:Wilhelm Wirtinger 356:Friedrich Hartogs 309:abelian functions 202: 128:solutions to the 126:square-integrable 109:in the variables 25:is the branch of 31363: 31306:Reinhardt domain 31277: 31261: 31227: 31213: 31203: 31201: 31176: 31174: 31149: 31147: 31122: 31110:Collected Papers 31104: 31075: 31047: 30980: 30962: 30950:"Stein manifold" 30944: 30926: 30908: 30890: 30872: 30860:"Coherent sheaf" 30854: 30836: 30818: 30800: 30782: 30764: 30746: 30728: 30710: 30683: 30662: 30627: 30601: 30575: 30562:10.1090/chel/340 30548: 30519: 30498: 30477: 30452: 30426: 30409: 30383: 30356: 30322: 30301: 30280: 30261: 30244: 30222: 30203: 30166: 30165: 30155: 30131: 30125: 30124: 30122: 30098: 30092: 30091: 30081: 30057: 30051: 30050: 30022: 30016: 30015: 29987: 29981: 29980: 29952: 29946: 29945: 29935: 29911: 29905: 29904: 29878: 29872: 29871: 29845: 29839: 29838: 29806: 29800: 29799: 29773: 29767: 29766: 29742: 29736: 29735: 29693: 29687: 29686: 29676: 29658: 29626: 29620: 29619: 29591: 29585: 29584: 29556: 29550: 29549: 29490: 29484: 29483: 29455: 29449: 29448: 29428: 29422: 29421: 29393: 29387: 29386: 29358: 29352: 29351: 29349: 29340:(3): 1185–1186. 29325: 29319: 29318: 29301: 29295: 29294: 29284: 29258: 29252: 29251: 29249: 29225: 29219: 29218: 29194: 29188: 29187: 29159: 29153: 29152: 29124: 29118: 29117: 29107: 29083: 29077: 29076: 29048: 29042: 29041: 29009: 29003: 29002: 29000: 28988: 28975: 28974: 28946: 28940: 28939: 28911: 28905: 28904: 28869: 28856: 28855: 28827: 28821: 28820: 28802: 28782: 28773: 28772: 28752: 28746: 28745: 28743: 28719: 28713: 28712: 28710: 28686: 28680: 28679: 28647: 28641: 28640: 28630: 28612: 28603: 28597: 28596: 28561: 28555: 28554: 28536: 28516: 28510: 28509: 28481: 28461: 28455: 28454: 28426: 28420: 28419: 28384: 28373: 28372: 28338: 28329: 28328: 28312: 28306: 28305: 28303: 28279: 28273: 28272: 28270: 28246: 28240: 28239: 28217: 28211: 28210: 28208: 28184: 28173: 28172: 28170: 28155:Acta Mathematica 28142: 28136: 28135: 28109: 28103: 28102: 28100: 28091: 28085: 28084: 28064: 28058: 28057: 28037: 28031: 28030: 28013: 27995: 27986: 27980: 27979: 27951: 27945: 27944: 27916: 27910: 27909: 27869: 27863: 27862: 27829: 27817: 27806: 27805: 27769: 27763: 27762: 27726: 27720: 27719: 27701: 27681: 27672: 27671: 27651: 27649: 27648: 27643: 27641: 27640: 27635: 27618: 27612: 27611: 27591: 27585: 27584: 27549: 27543: 27542: 27515: 27509: 27508: 27499: 27471: 27465: 27464: 27447: 27441: 27440: 27401: 27395: 27394: 27392: 27368: 27362: 27361: 27351: 27325: 27319: 27318: 27286: 27280: 27279: 27259: 27250: 27249: 27247: 27245:10.5802/aif.3226 27238:(7): 2811–2818. 27227: 27225: 27224: 27219: 27217: 27201: 27195: 27194: 27177: 27171: 27170: 27168: 27144: 27135: 27134: 27125: 27105: 27092: 27091: 27089: 27065: 27059: 27058: 27056: 27032: 27026: 27025: 27005: 26999: 26998: 26978: 26972: 26967: 26954: 26953: 26951: 26927: 26921: 26920: 26918: 26894: 26888: 26887: 26859: 26853: 26852: 26822: 26813: 26812: 26807:. Archived from 26800: 26791: 26790: 26770: 26764: 26763: 26736: 26730: 26729: 26693: 26678: 26677: 26665: 26659: 26658: 26634: 26625: 26624: 26622: 26598: 26585: 26584: 26583: 26563: 26562: 26542: 26533: 26532: 26526: 26517: 26515: 26491: 26480: 26479: 26471: 26465: 26464: 26462: 26453: 26447: 26446: 26428: 26408: 26395: 26394: 26385: 26365: 26359: 26358: 26349: 26329: 26318: 26317: 26297:, First Series, 26284: 26275: 26274: 26272: 26248: 26242: 26241: 26231: 26207: 26201: 26200: 26190: 26166: 26160: 26159: 26137: 26131: 26130: 26128: 26104: 26098: 26097: 26087: 26077: 26053: 26047: 26046: 26036: 26027:(6): 1244–1249. 26010: 26004: 26003: 26001: 25989: 25980: 25979: 25953: 25944: 25943: 25915: 25909: 25908: 25880: 25874: 25873: 25863: 25839: 25833: 25832: 25804: 25798: 25797: 25765: 25759: 25758: 25748: 25724: 25718: 25717: 25691: 25682: 25681: 25661: 25655: 25654: 25637: 25631: 25630: 25628: 25604: 25598: 25597: 25571: 25542: 25531: 25530: 25504: 25498: 25497: 25469: 25458: 25457: 25431: 25407: 25392: 25391: 25381: 25357: 25334: 25333: 25323: 25299: 25290: 25289: 25269: 25263: 25262: 25252: 25237:Acta Mathematica 25232: 25230: 25229: 25224: 25222: 25221: 25213: 25199: 25186:Inline citations 25174: 25172: 25170: 25169: 25164: 25162: 25150: 25148: 25147: 25142: 25140: 25139: 25134: 25120: 25114: 25110: 25104: 25101:Oka-Weil theorem 25093: 25087: 25085: 25083: 25082: 25077: 25075: 25074: 25069: 25056: 25054: 25053: 25048: 25046: 25045: 25040: 25031: 25030: 25025: 25012: 25006: 25003: 24997: 24994: 24988: 24985: 24979: 24976: 24970: 24967: 24961: 24958: 24952: 24946: 24940: 24937: 24931: 24927: 24921: 24918: 24912: 24909: 24903: 24897: 24895: 24894: 24889: 24887: 24886: 24881: 24867: 24861: 24852:The idea of the 24850: 24844: 24841: 24835: 24832: 24826: 24819: 24813: 24809: 24803: 24801: 24799: 24798: 24793: 24791: 24790: 24769: 24763: 24756: 24750: 24747: 24741: 24730: 24724: 24711: 24695:Oka–Weil theorem 24665:Complex geometry 24660:Bicomplex number 24649: 24647: 24646: 24641: 24639: 24638: 24633: 24559: 24557: 24556: 24551: 24549: 24548: 24543: 24491: 24489: 24488: 24483: 24481: 24480: 24475: 24463: 24462: 24457: 24452: 24442: 24440: 24439: 24434: 24432: 24391:affine varieties 24380:sheaf cohomology 24354: 24352: 24351: 24346: 24326: 24325: 24306: 24304: 24303: 24298: 24275: 24274: 24244: 24242: 24241: 24236: 24225: 24224: 24209: 24208: 24173: 24171: 24170: 24165: 24109: 24107: 24106: 24101: 24073: 24071: 24070: 24065: 24020: 24018: 24017: 24012: 24001: 24000: 23992: 23968: 23966: 23965: 23960: 23947:plurisubharmonic 23922: 23920: 23919: 23914: 23847: 23845: 23844: 23839: 23837: 23836: 23822: 23791: 23789: 23788: 23783: 23781: 23780: 23775: 23759: 23757: 23756: 23751: 23749: 23748: 23743: 23721: 23719: 23718: 23713: 23679: 23675: 23674: 23659: 23658: 23618: 23616: 23615: 23610: 23608: 23607: 23591: 23589: 23588: 23583: 23578: 23577: 23549: 23547: 23546: 23541: 23539: 23538: 23533: 23516: 23514: 23513: 23508: 23506: 23505: 23487: 23486: 23470: 23468: 23467: 23462: 23454: 23453: 23410: 23408: 23407: 23402: 23400: 23399: 23389: 23388: 23339: 23337: 23336: 23331: 23287: 23285: 23284: 23279: 23261: 23259: 23258: 23253: 23232: 23230: 23229: 23224: 23203: 23201: 23200: 23195: 23133: 23131: 23130: 23125: 23092: 23090: 23089: 23084: 23082: 23081: 23076: 23039: 23037: 23036: 23031: 22991: 22989: 22988: 22983: 22972: 22958: 22957: 22941: 22939: 22938: 22933: 22922: 22921: 22916: 22915: 22899: 22898: 22883: 22882: 22877: 22876: 22860: 22859: 22836: 22834: 22833: 22828: 22823: 22822: 22817: 22816: 22800: 22799: 22784: 22770: 22769: 22753: 22748: 22743: 22742: 22726: 22725: 22710: 22709: 22704: 22703: 22687: 22686: 22663: 22661: 22660: 22655: 22643: 22638: 22633: 22632: 22616: 22615: 22552: 22550: 22549: 22544: 22533: 22532: 22513: 22511: 22510: 22505: 22466: 22464: 22463: 22458: 22447: 22446: 22423: 22421: 22420: 22415: 22377: 22375: 22374: 22369: 22364: 22360: 22350: 22349: 22332: 22328: 22311: 22305: 22287: 22270: 22245: 22244: 22236: 22221:, the so-called 22220: 22218: 22217: 22212: 22175: 22173: 22172: 22167: 22156: 22155: 22142: 22140: 22139: 22134: 22097:complex manifold 22094: 22092: 22091: 22086: 22084: 22083: 22078: 22060:affine varieties 22034: 22032: 22031: 22026: 22024: 22023: 22018: 21997: 21995: 21994: 21989: 21987: 21986: 21981: 21968: 21966: 21965: 21960: 21958: 21957: 21952: 21931: 21929: 21928: 21923: 21921: 21920: 21915: 21902: 21900: 21899: 21894: 21892: 21891: 21883: 21854: 21852: 21851: 21846: 21844: 21811: 21809: 21808: 21803: 21792: 21778: 21777: 21761: 21759: 21758: 21753: 21735: 21733: 21732: 21727: 21716: 21702: 21701: 21678: 21676: 21675: 21670: 21662: 21648: 21647: 21632: 21618: 21617: 21593: 21592: 21587: 21572: 21571: 21556: 21542: 21541: 21522: 21520: 21519: 21514: 21509: 21495: 21494: 21474: 21472: 21471: 21466: 21464: 21440: 21438: 21437: 21432: 21424: 21423: 21418: 21403: 21402: 21394: 21364: 21362: 21361: 21356: 21351: 21337: 21336: 21320: 21318: 21317: 21312: 21310: 21309: 21304: 21291: 21289: 21288: 21283: 21278: 21277: 21272: 21257: 21256: 21238: 21236: 21235: 21230: 21219: 21218: 21213: 21198: 21197: 21178: 21176: 21175: 21170: 21165: 21164: 21159: 21144: 21143: 21128: 21127: 21122: 21116: 21111: 21110: 21105: 21090: 21089: 21080: 21070: 21065: 21064: 21059: 21044: 21043: 21021: 21019: 21018: 21013: 21005: 21004: 20999: 20993: 20988: 20987: 20982: 20967: 20966: 20957: 20947: 20942: 20941: 20936: 20921: 20920: 20901: 20899: 20898: 20893: 20881: 20879: 20878: 20873: 20871: 20870: 20865: 20859: 20854: 20853: 20848: 20831: 20829: 20828: 20823: 20821: 20820: 20815: 20802: 20800: 20799: 20794: 20792: 20791: 20786: 20770: 20768: 20767: 20762: 20760: 20759: 20750: 20731: 20729: 20728: 20723: 20711: 20709: 20708: 20703: 20701: 20700: 20691: 20686: 20685: 20638: 20636: 20635: 20630: 20625: 20611: 20610: 20595: 20590: 20585: 20571: 20570: 20561: 20551: 20546: 20532: 20531: 20505: 20503: 20502: 20497: 20489: 20484: 20479: 20465: 20464: 20455: 20445: 20440: 20426: 20425: 20387: 20385: 20384: 20379: 20367: 20365: 20364: 20359: 20357: 20356: 20336: 20334: 20333: 20328: 20326: 20325: 20299: 20297: 20296: 20291: 20279: 20277: 20276: 20271: 20269: 20268: 20256: 20255: 20239: 20237: 20236: 20231: 20229: 20228: 20212: 20210: 20209: 20204: 20202: 20201: 20132: 20130: 20129: 20124: 20122: 20121: 20116: 20115: 20094: 20092: 20091: 20086: 20084: 20083: 20078: 20065: 20063: 20062: 20057: 20046: 20045: 20033:the ideal sheaf 20030: 20028: 20027: 20022: 20017: 20016: 20011: 20010: 19987: 19985: 19984: 19979: 19977: 19976: 19971: 19970: 19956: 19954: 19953: 19948: 19946: 19945: 19940: 19923: 19921: 19920: 19915: 19913: 19912: 19911: 19910: 19905: 19898: 19897: 19887: 19886: 19860: 19858: 19857: 19852: 19850: 19849: 19844: 19843: 19829: 19827: 19826: 19821: 19819: 19818: 19805: 19803: 19802: 19797: 19789: 19788: 19783: 19777: 19776: 19771: 19770: 19760: 19759: 19754: 19748: 19747: 19742: 19741: 19731: 19730: 19725: 19719: 19718: 19713: 19712: 19681: 19679: 19678: 19673: 19671: 19670: 19665: 19664: 19646: 19644: 19643: 19638: 19626: 19624: 19623: 19618: 19616: 19615: 19610: 19609: 19595: 19593: 19592: 19587: 19585: 19584: 19579: 19573: 19572: 19563: 19562: 19557: 19550: 19542: 19537: 19536: 19516: 19514: 19513: 19508: 19490: 19488: 19487: 19482: 19461: 19459: 19458: 19453: 19441: 19439: 19438: 19433: 19431: 19430: 19425: 19419: 19418: 19409: 19408: 19403: 19396: 19388: 19383: 19382: 19368: 19366: 19365: 19360: 19348: 19346: 19345: 19340: 19325: 19323: 19322: 19317: 19305: 19303: 19302: 19297: 19295: 19294: 19289: 19288: 19270: 19268: 19267: 19262: 19260: 19259: 19243: 19241: 19240: 19235: 19233: 19232: 19219: 19217: 19216: 19211: 19206: 19205: 19200: 19199: 19169: 19167: 19166: 19161: 19149: 19147: 19146: 19141: 19126: 19124: 19123: 19118: 19110: 19109: 19104: 19098: 19097: 19088: 19087: 19082: 19075: 19067: 19062: 19061: 19051: 19050: 19045: 19038: 19030: 19025: 19024: 19004: 19002: 19001: 18996: 18984: 18982: 18981: 18976: 18960: 18958: 18957: 18952: 18950: 18949: 18944: 18943: 18929: 18927: 18926: 18921: 18919: 18918: 18905: 18903: 18902: 18897: 18892: 18891: 18886: 18885: 18826: 18824: 18823: 18818: 18791: 18789: 18788: 18783: 18763: 18746: 18722: 18720: 18719: 18714: 18697: 18686: 18647: 18645: 18644: 18639: 18622: 18589: 18587: 18586: 18581: 18576: 18552: 18550: 18549: 18544: 18504: 18502: 18501: 18496: 18484: 18482: 18481: 18476: 18464: 18462: 18461: 18456: 18432: 18430: 18429: 18424: 18382: 18380: 18379: 18374: 18372: 18371: 18363: 18349: 18347: 18346: 18341: 18339: 18338: 18333: 18312: 18310: 18309: 18304: 18286: 18284: 18283: 18278: 18260: 18258: 18257: 18252: 18234: 18232: 18231: 18226: 18193:is pseudoconvex. 18159: 18157: 18156: 18151: 18149: 18148: 18143: 18116: 18114: 18113: 18108: 18096: 18094: 18093: 18088: 18076: 18074: 18073: 18068: 18066: 18065: 18060: 18039: 18037: 18036: 18031: 18029: 18028: 18023: 18022: 18008: 18006: 18005: 18000: 17989: 17988: 17969: 17967: 17966: 17961: 17943: 17941: 17940: 17935: 17903: 17901: 17900: 17895: 17865: 17863: 17862: 17857: 17837:For every point 17828: 17826: 17825: 17820: 17818: 17817: 17812: 17791: 17789: 17788: 17783: 17744: 17742: 17741: 17736: 17709: 17707: 17706: 17701: 17682: 17656: 17654: 17653: 17648: 17637: 17629: 17606: 17605: 17593: 17592: 17554: 17552: 17551: 17546: 17535: 17527: 17504: 17503: 17491: 17490: 17456: 17454: 17453: 17448: 17436: 17434: 17433: 17428: 17426: 17424: 17416: 17415: 17414: 17401: 17388: 17386: 17385: 17380: 17372: 17364: 17352: 17350: 17349: 17344: 17318: 17310: 17292: 17290: 17289: 17284: 17267: 17266: 17254: 17253: 17219:Oka pseudoconvex 17210: 17208: 17207: 17202: 17190: 17188: 17187: 17182: 17166: 17164: 17163: 17158: 17146: 17144: 17143: 17138: 17136: 17135: 17122:analytic variety 17115: 17113: 17112: 17107: 17079: 17077: 17076: 17071: 17069: 17068: 17063: 17062: 17048: 17046: 17045: 17040: 17025: 17023: 17022: 17017: 17012: 17011: 17001: 16996: 16960: 16958: 16957: 16952: 16950: 16949: 16944: 16943: 16929: 16927: 16926: 16921: 16913: 16912: 16881: 16879: 16878: 16873: 16871: 16870: 16865: 16864: 16851:does not have a 16843: 16841: 16840: 16835: 16827: 16826: 16821: 16820: 16811: 16808: 16807: 16798: 16796: 16795: 16794: 16789: 16788: 16779: 16772: 16771: 16758: 16745: 16744: 16734: 16731: 16726: 16681: 16679: 16678: 16673: 16665: 16664: 16655: 16653: 16652: 16651: 16638: 16621: 16618: 16613: 16567: 16565: 16564: 16559: 16547: 16545: 16544: 16539: 16514: 16512: 16511: 16506: 16473: 16471: 16470: 16465: 16435: 16433: 16432: 16427: 16425: 16424: 16419: 16418: 16404: 16402: 16401: 16396: 16394: 16386: 16385: 16380: 16357: 16355: 16354: 16349: 16347: 16346: 16341: 16340: 16312: 16310: 16309: 16304: 16289: 16287: 16286: 16281: 16270: 16269: 16264: 16263: 16244: 16241: 16202: 16200: 16199: 16194: 16183: 16182: 16177: 16176: 16157: 16154: 16131: 16129: 16128: 16123: 16112: 16111: 16106: 16105: 16086: 16083: 16056: 16054: 16053: 16048: 15996: 15994: 15993: 15988: 15962: 15960: 15959: 15954: 15952: 15951: 15946: 15945: 15913: 15911: 15910: 15905: 15903: 15902: 15897: 15896: 15866: 15864: 15863: 15858: 15853: 15852: 15844: 15838: 15830: 15817: 15815: 15814: 15809: 15807: 15806: 15801: 15800: 15787:Equivalently, a 15783: 15781: 15780: 15775: 15773: 15771: 15770: 15769: 15764: 15763: 15755: 15747: 15746: 15733: 15729: 15728: 15718: 15713: 15712: 15694: 15693: 15675: 15674: 15658:hermitian matrix 15655: 15653: 15652: 15647: 15635: 15633: 15632: 15627: 15625: 15624: 15619: 15618: 15604: 15602: 15601: 15596: 15585:. Therefore, if 15584: 15582: 15581: 15576: 15568: 15564: 15562: 15561: 15553: 15540: 15536: 15535: 15525: 15499: 15497: 15496: 15491: 15461: 15459: 15458: 15453: 15451: 15430: 15428: 15427: 15422: 15405: 15372: 15370: 15369: 15364: 15332: 15330: 15329: 15324: 15307: 15278: 15276: 15275: 15270: 15250: 15248: 15247: 15242: 15213: 15188: 15186: 15185: 15180: 15140: 15138: 15137: 15132: 15130: 15129: 15124: 15099: 15097: 15096: 15091: 15089: 15088: 15083: 15071: 15027:plurisubharmonic 15021: 15019: 15018: 15013: 15011: 15010: 15005: 15004: 14979: 14977: 14976: 14971: 14951: 14950: 14898: 14896: 14895: 14890: 14885: 14884: 14872: 14851: 14849: 14848: 14843: 14838: 14837: 14819: 14814: 14813: 14804: 14792: 14790: 14789: 14784: 14776: 14775: 14759: 14757: 14756: 14751: 14749: 14748: 14743: 14730: 14728: 14727: 14722: 14702: 14700: 14699: 14694: 14692: 14664: 14662: 14661: 14656: 14651: 14650: 14645: 14623: 14621: 14620: 14615: 14607: 14606: 14601: 14592: 14591: 14575: 14573: 14572: 14567: 14562: 14561: 14543: 14540: 14523: 14521: 14520: 14515: 14510: 14509: 14497: 14496: 14491: 14490: 14482: 14475: 14472: 14464: 14463: 14445: 14442: 14425: 14423: 14422: 14417: 14396: 14394: 14393: 14388: 14361:is the union of 14360: 14358: 14357: 14352: 14350: 14349: 14344: 14343: 14335: 14324: 14322: 14321: 14316: 14304: 14302: 14301: 14296: 14274:holomorph-convex 14267: 14265: 14264: 14259: 14257: 14256: 14251: 14250: 14242: 14221: 14219: 14218: 14213: 14194: 14192: 14191: 14186: 14175: 14174: 14154: 14152: 14151: 14146: 14141: 14137: 14124: 14123: 14111: 14108: 14106: 14089: 14083: 14065: 14048: 14023: 14022: 14017: 14016: 14008: 13986: 13984: 13983: 13978: 13956: 13954: 13953: 13948: 13937: 13936: 13919: 13917: 13916: 13911: 13899: 13897: 13896: 13891: 13879: 13877: 13876: 13871: 13869: 13868: 13863: 13835: 13833: 13832: 13827: 13802: 13800: 13799: 13794: 13792: 13791: 13779: 13778: 13762: 13760: 13759: 13754: 13752: 13751: 13735: 13733: 13732: 13727: 13725: 13724: 13701: 13699: 13698: 13693: 13691: 13690: 13680: 13675: 13647: 13645: 13644: 13639: 13631: 13630: 13618: 13617: 13598: 13596: 13595: 13590: 13588: 13587: 13577: 13572: 13544: 13542: 13541: 13536: 13534: 13533: 13515: 13514: 13488:natural boundary 13477: 13475: 13474: 13469: 13467: 13449: 13447: 13446: 13441: 13416: 13414: 13413: 13408: 13374: 13372: 13371: 13366: 13342: 13340: 13339: 13334: 13316: 13314: 13313: 13308: 13306: 13305: 13300: 13281: 13279: 13278: 13273: 13251: 13249: 13248: 13243: 13241: 13240: 13235: 13211: 13209: 13208: 13203: 13184: 13183: 13182: 13177: 13134: 13132: 13131: 13126: 13107: 13105: 13104: 13099: 13097: 13096: 13091: 13069:When a function 13061: 13059: 13058: 13053: 13024:sheaf cohomology 13011: 13009: 13008: 13003: 13001: 13000: 12995: 12982: 12980: 12979: 12974: 12972: 12971: 12966: 12949: 12947: 12946: 12941: 12939: 12938: 12933: 12910: 12908: 12907: 12902: 12900: 12899: 12884: 12883: 12861: 12859: 12858: 12853: 12841: 12839: 12838: 12833: 12822: 12821: 12809: 12808: 12790: 12789: 12777: 12776: 12760: 12758: 12757: 12752: 12750: 12749: 12744: 12735: 12734: 12729: 12710: 12708: 12707: 12702: 12700: 12699: 12683: 12681: 12680: 12675: 12673: 12672: 12647:Toshikazu Sunada 12642:Sunada's results 12636: 12634: 12633: 12628: 12592: 12591: 12583: 12581: 12572: 12564: 12563: 12558: 12549: 12542: 12538: 12537: 12532: 12496: 12494: 12493: 12488: 12477: 12476: 12471: 12462: 12454: 12453: 12448: 12439: 12432: 12428: 12427: 12422: 12386: 12384: 12383: 12378: 12367: 12359: 12352: 12342: 12334: 12327: 12323: 12322: 12317: 12265: 12263: 12262: 12257: 12255: 12254: 12238: 12236: 12235: 12230: 12228: 12227: 12211: 12209: 12208: 12203: 12201: 12200: 12184: 12182: 12181: 12176: 12174: 12173: 12157: 12155: 12154: 12149: 12147: 12146: 12130: 12128: 12127: 12122: 12120: 12119: 12081: 12060: 12053: 12051: 12050: 12045: 12043: 12042: 12037: 12024: 12020: 11993: 11991: 11990: 11985: 11960: 11956: 11951: 11950: 11941: 11922: 11916: 11909: 11904: 11903: 11894: 11886: 11885: 11870: 11869: 11857: 11856: 11832: 11831: 11810: 11808: 11807: 11802: 11778: 11776: 11775: 11770: 11759: 11754: 11753: 11744: 11730: 11725: 11724: 11715: 11707: 11706: 11701: 11683: 11682: 11662: 11660: 11659: 11654: 11652: 11651: 11646: 11620: 11618: 11617: 11612: 11594: 11592: 11591: 11586: 11584: 11583: 11565: 11564: 11548: 11546: 11545: 11540: 11536: 11535: 11534: 11533: 11532: 11518: 11517: 11505: 11504: 11489: 11488: 11487: 11486: 11472: 11471: 11459: 11458: 11446: 11445: 11444: 11443: 11425: 11424: 11409: 11404: 11397: 11396: 11378: 11377: 11356: 11354: 11353: 11348: 11346: 11345: 11340: 11324: 11322: 11321: 11316: 11314: 11313: 11308: 11288: 11286: 11285: 11280: 11275: 11270: 11269: 11260: 11240: 11235: 11234: 11225: 11171: 11169: 11168: 11163: 11152: 11151: 11133: 11132: 11111: 11110: 11092: 11091: 11067: 11066: 11047: 11045: 11044: 11039: 11034: 11033: 10983:simply connected 10965: 10963: 10962: 10957: 10952: 10948: 10924: 10920: 10916: 10915: 10914: 10901: 10896: 10879: 10875: 10874: 10873: 10861: 10860: 10840: 10839: 10821: 10820: 10787: 10785: 10784: 10779: 10771: 10770: 10743: 10741: 10740: 10735: 10730: 10726: 10705: 10701: 10700: 10688: 10684: 10677: 10676: 10644: 10643: 10625: 10624: 10591: 10589: 10588: 10583: 10575: 10574: 10555: 10553: 10552: 10547: 10545: 10541: 10537: 10536: 10523: 10518: 10506: 10505: 10504: 10503: 10478: 10474: 10473: 10472: 10460: 10459: 10434: 10432: 10431: 10426: 10396: 10395: 10369: 10367: 10366: 10361: 10356: 10352: 10328: 10324: 10320: 10319: 10318: 10305: 10300: 10283: 10279: 10278: 10277: 10265: 10264: 10244: 10243: 10225: 10224: 10191: 10189: 10188: 10183: 10171: 10166: 10147: 10142: 10127: 10126: 10110: 10108: 10107: 10102: 10100: 10099: 10094: 10082: 10081: 10063: 10062: 10037: 10035: 10034: 10029: 10011: 10009: 10008: 10003: 10001: 10000: 9995: 9969:Reinhardt domain 9952: 9950: 9949: 9944: 9925: 9923: 9922: 9917: 9902: 9900: 9899: 9894: 9883: 9882: 9863: 9861: 9860: 9855: 9844: 9843: 9821:over the domain 9820: 9818: 9817: 9812: 9792: 9790: 9789: 9784: 9745: 9743: 9742: 9737: 9735: 9734: 9729: 9717: 9716: 9711: 9695: 9693: 9692: 9687: 9665: 9663: 9662: 9657: 9645: 9643: 9642: 9637: 9618: 9609: 9591: 9589: 9588: 9583: 9572: 9571: 9558: 9556: 9555: 9550: 9539: 9538: 9519: 9517: 9516: 9511: 9500: 9499: 9480: 9478: 9477: 9472: 9470: 9469: 9464: 9419: 9417: 9416: 9411: 9369: 9367: 9366: 9361: 9349: 9347: 9346: 9341: 9327: 9326: 9307: 9305: 9304: 9299: 9275: 9273: 9272: 9267: 9265: 9264: 9259: 9208: 9206: 9205: 9200: 9192: 9191: 9175: 9173: 9172: 9167: 9165: 9164: 9159: 9146: 9144: 9143: 9138: 9136: 9135: 9120: 9119: 9107: 9106: 9084: 9082: 9081: 9076: 9065: 9064: 9052: 9051: 9017: 9015: 9014: 9009: 9007: 9006: 9001: 8981:identity theorem 8975:Identity theorem 8970: 8968: 8967: 8962: 8926: 8925: 8880: 8878: 8877: 8872: 8833: 8825: 8823: 8822: 8780: 8779: 8743: 8741: 8740: 8735: 8733: 8732: 8717: 8716: 8711: 8703: 8688: 8687: 8666: 8665: 8650: 8649: 8644: 8636: 8626: 8625: 8620: 8612: 8606: 8605: 8600: 8592: 8585: 8564: 8562: 8561: 8560: 8552: 8537: 8528: 8526: 8524: 8523: 8522: 8500: 8480: 8447: 8445: 8444: 8439: 8409: 8407: 8406: 8401: 8385: 8383: 8382: 8377: 8350: 8348: 8347: 8342: 8340: 8339: 8334: 8317: 8315: 8314: 8309: 8285: 8283: 8282: 8277: 8265: 8263: 8262: 8257: 8242: 8240: 8239: 8234: 8232: 8231: 8226: 8190: 8188: 8187: 8182: 8180: 8179: 8163: 8147: 8145: 8144: 8139: 8137: 8136: 8120: 8104: 8102: 8101: 8096: 8090: 8078: 8070: 8058: 8039: 8037: 8036: 8031: 8029: 8016: 8015: 7997: 7996: 7982: 7981: 7979: 7978: 7966: 7961: 7960: 7948: 7944: 7933: 7926: 7925: 7915: 7902: 7895: 7894: 7884: 7872: 7869: 7865: 7864: 7849: 7848: 7838: 7830: 7829: 7789: 7788: 7787: 7786: 7774: 7769: 7768: 7759: 7749: 7747: 7733: 7731: 7729: 7718: 7715: 7710: 7689: 7685: 7684: 7672: 7671: 7659: 7658: 7647: 7643: 7642: 7640: 7626: 7624: 7622: 7621: 7620: 7607: 7606: 7597: 7568: 7567: 7566: 7565: 7553: 7548: 7547: 7538: 7528: 7526: 7525: 7524: 7499: 7497: 7495: 7484: 7481: 7476: 7428: 7426: 7425: 7420: 7418: 7414: 7392: 7389: 7384: 7383: 7371: 7363: 7355: 7354: 7342: 7341: 7336: 7324: 7323: 7305: 7304: 7292: 7291: 7258: 7256: 7255: 7250: 7217: 7215: 7214: 7209: 7207: 7203: 7181: 7178: 7173: 7172: 7160: 7155: 7154: 7142: 7141: 7132: 7124: 7123: 7118: 7106: 7105: 7087: 7086: 7074: 7073: 7043: 7041: 7040: 7035: 7033: 7029: 7007: 7004: 6999: 6998: 6986: 6981: 6980: 6968: 6967: 6958: 6950: 6949: 6944: 6932: 6931: 6913: 6912: 6900: 6899: 6869: 6867: 6866: 6861: 6857: 6856: 6855: 6854: 6853: 6839: 6838: 6826: 6825: 6810: 6809: 6808: 6807: 6793: 6792: 6780: 6779: 6767: 6766: 6765: 6764: 6746: 6745: 6730: 6725: 6718: 6717: 6699: 6698: 6677: 6675: 6674: 6669: 6635: 6634: 6633: 6606: 6604: 6603: 6598: 6596: 6594: 6593: 6592: 6591: 6590: 6580: 6579: 6578: 6561: 6560: 6559: 6558: 6548: 6547: 6546: 6531: 6530: 6529: 6520: 6519: 6518: 6517: 6499: 6498: 6483: 6480: 6475: 6457: 6455: 6454: 6453: 6452: 6451: 6441: 6440: 6439: 6422: 6421: 6420: 6419: 6409: 6408: 6407: 6392: 6388: 6387: 6386: 6385: 6367: 6366: 6351: 6330: 6328: 6327: 6322: 6320: 6319: 6299: 6297: 6296: 6291: 6289: 6288: 6264: 6262: 6261: 6256: 6254: 6253: 6235: 6234: 6210: 6208: 6207: 6202: 6195: 6194: 6193: 6192: 6191: 6177: 6176: 6164: 6163: 6148: 6147: 6146: 6145: 6131: 6130: 6118: 6117: 6105: 6104: 6103: 6102: 6084: 6083: 6068: 6063: 6056: 6055: 6037: 6036: 5993: 5991: 5990: 5985: 5964: 5962: 5961: 5956: 5954: 5953: 5948: 5930: 5929: 5911: 5910: 5875: 5873: 5872: 5867: 5865: 5861: 5860: 5845: 5844: 5831: 5829: 5828: 5827: 5820: 5819: 5805: 5804: 5792: 5791: 5776: 5775: 5768: 5767: 5753: 5752: 5740: 5739: 5726: 5722: 5721: 5703: 5702: 5686: 5684: 5683: 5682: 5681: 5661: 5660: 5659: 5658: 5641: 5639: 5638: 5637: 5612: 5607: 5606: 5605: 5604: 5592: 5591: 5576: 5567: 5566: 5565: 5564: 5563: 5549: 5548: 5536: 5535: 5520: 5519: 5518: 5517: 5503: 5502: 5490: 5489: 5477: 5476: 5475: 5474: 5456: 5455: 5440: 5435: 5428: 5427: 5409: 5408: 5378: 5364: 5362: 5361: 5356: 5330: 5327: 5322: 5321: 5309: 5304: 5303: 5291: 5290: 5281: 5273: 5272: 5267: 5255: 5254: 5236: 5235: 5223: 5222: 5175: 5173: 5172: 5167: 5165: 5163: 5162: 5161: 5160: 5159: 5149: 5148: 5147: 5133: 5132: 5131: 5130: 5120: 5119: 5118: 5106: 5102: 5101: 5089: 5088: 5075: 5070: 5066: 5064: 5063: 5062: 5061: 5060: 5050: 5049: 5048: 5031: 5030: 5029: 5028: 5018: 5017: 5016: 5001: 4997: 4996: 4978: 4977: 4965: 4964: 4949: 4948: 4947: 4946: 4928: 4927: 4912: 4896: 4894: 4893: 4888: 4886: 4878: 4870: 4858: 4856: 4855: 4850: 4848: 4844: 4822: 4819: 4814: 4813: 4801: 4796: 4795: 4783: 4782: 4773: 4765: 4764: 4759: 4747: 4746: 4728: 4727: 4715: 4714: 4677: 4675: 4674: 4669: 4667: 4666: 4661: 4660: 4636: 4627: 4625: 4624: 4619: 4614: 4613: 4598: 4597: 4584: 4582: 4581: 4580: 4573: 4572: 4558: 4557: 4545: 4544: 4529: 4528: 4521: 4520: 4506: 4505: 4493: 4492: 4479: 4475: 4474: 4456: 4455: 4439: 4437: 4436: 4435: 4434: 4414: 4413: 4412: 4411: 4394: 4392: 4391: 4390: 4368: 4364: 4363: 4351: 4350: 4340: 4335: 4333: 4332: 4331: 4330: 4329: 4319: 4318: 4317: 4300: 4299: 4298: 4297: 4287: 4286: 4285: 4270: 4266: 4265: 4247: 4246: 4234: 4233: 4218: 4217: 4216: 4215: 4197: 4196: 4181: 4171: 4149: 4140: 4138: 4137: 4132: 4130: 4129: 4114: 4113: 4100: 4098: 4094: 4093: 4081: 4080: 4062: 4061: 4049: 4048: 4035: 4031: 4030: 4012: 4011: 3995: 3993: 3992: 3991: 3990: 3970: 3969: 3968: 3967: 3950: 3948: 3947: 3946: 3921: 3913: 3912: 3894: 3893: 3870: 3854: 3852: 3851: 3846: 3828: 3826: 3825: 3820: 3818: 3814: 3813: 3800: 3798: 3794: 3793: 3781: 3780: 3762: 3761: 3749: 3748: 3733: 3732: 3720: 3719: 3706: 3702: 3701: 3683: 3682: 3670: 3669: 3653: 3651: 3650: 3649: 3648: 3631: 3630: 3617: 3616: 3615: 3614: 3594: 3593: 3580: 3579: 3578: 3577: 3560: 3558: 3557: 3556: 3531: 3523: 3519: 3518: 3505: 3503: 3499: 3498: 3486: 3485: 3470: 3469: 3457: 3456: 3443: 3439: 3438: 3420: 3419: 3407: 3406: 3394: 3393: 3377: 3375: 3374: 3373: 3372: 3355: 3354: 3341: 3340: 3339: 3338: 3321: 3319: 3318: 3317: 3292: 3284: 3280: 3279: 3266: 3264: 3263: 3262: 3250: 3249: 3239: 3235: 3234: 3216: 3215: 3203: 3202: 3186: 3184: 3183: 3182: 3181: 3164: 3162: 3148: 3136: 3135: 3117: 3116: 3083: 3081: 3080: 3075: 3072: 3067: 3039: 3037: 3036: 3031: 3029: 3025: 3003: 3000: 2998: 2997: 2985: 2981: 2980: 2979: 2967: 2966: 2949: 2948: 2943: 2931: 2930: 2912: 2911: 2899: 2898: 2857: 2849: 2840: 2838: 2837: 2832: 2830: 2829: 2828: 2810: 2809: 2797: 2796: 2783: 2775: 2766: 2764: 2763: 2758: 2756: 2748: 2736: 2734: 2733: 2728: 2720: 2715: 2714: 2713: 2695: 2694: 2682: 2681: 2671: 2662: 2660: 2659: 2654: 2652: 2647: 2646: 2645: 2627: 2626: 2614: 2613: 2603: 2594: 2592: 2591: 2586: 2584: 2583: 2567: 2565: 2564: 2559: 2557: 2556: 2540: 2538: 2537: 2532: 2496: 2494: 2493: 2488: 2486: 2485: 2480: 2479: 2461: 2459: 2458: 2453: 2451: 2450: 2434: 2432: 2431: 2426: 2396: 2394: 2393: 2388: 2377: 2376: 2368: 2354: 2352: 2351: 2346: 2335: 2333: 2332: 2327: 2326: 2317: 2311: 2303: 2254: 2252: 2251: 2246: 2244: 2242: 2241: 2240: 2227: 2219: 2211: 2209: 2208: 2207: 2194: 2186: 2183: 2180: 2177: 2175: 2174: 2173: 2160: 2152: 2147: 2145: 2144: 2143: 2130: 2122: 2079: 2077: 2076: 2071: 2059: 2057: 2056: 2051: 2039: 2037: 2036: 2031: 2019: 2017: 2016: 2011: 2009: 2001: 2000: 1995: 1974: 1972: 1971: 1966: 1955: 1953: 1945: 1937: 1920: 1918: 1910: 1902: 1899: 1896: 1884: 1882: 1874: 1866: 1852: 1850: 1842: 1834: 1825: 1823: 1822: 1817: 1801: 1799: 1798: 1793: 1781: 1779: 1778: 1773: 1761: 1759: 1758: 1753: 1751: 1733: 1731: 1730: 1725: 1723: 1715: 1676:Hartog's theorem 1659: 1657: 1656: 1651: 1646: 1645: 1627: 1626: 1602: 1601: 1577: 1576: 1529: 1527: 1526: 1521: 1441: 1439: 1438: 1433: 1431: 1423: 1422: 1417: 1398: 1396: 1395: 1390: 1372: 1370: 1369: 1364: 1362: 1350: 1348: 1347: 1342: 1340: 1339: 1334: 1298: 1296: 1295: 1290: 1288: 1287: 1282: 1269: 1267: 1266: 1261: 1259: 1258: 1253: 1217: 1215: 1214: 1209: 1204: 1203: 1198: 1189: 1181: 1180: 1168: 1167: 1142: 1140: 1139: 1134: 1129: 1128: 1077: 1056:thought of as a 1044: 1034:) which defines 1033: 1020: 1006: 995: 993: 992: 987: 985: 984: 976: 960: 958: 957: 952: 950: 949: 944: 927: 925: 924: 919: 917: 905: 892: 879: 877: 876: 871: 869: 868: 863: 846: 844: 843: 838: 836: 835: 830: 817: 815: 814: 809: 807: 806: 801: 788: 786: 785: 780: 778: 766: 756: 754: 753: 748: 746: 745: 740: 705:theory, and the 692:symplectic group 689: 677:Weil restriction 673:algebraic groups 653:special function 613:Kunihiko Kodaira 577:sheaf cohomology 562: 560: 559: 554: 552: 551: 546: 529: 522: 520: 519: 514: 512: 491:Reinhold Remmert 476:residue calculus 465: 454: 447: 445: 444: 439: 437: 429: 428: 423: 395:Francesco Severi 368: 290: 288: 287: 282: 280: 279: 274: 261: 259: 258: 253: 251: 250: 245: 221: 219: 218: 213: 200: 196: 195: 190: 167: 165: 164: 159: 157: 131: 115: 103:complex analytic 94: 64: 60: 58: 57: 52: 50: 49: 44: 31371: 31370: 31366: 31365: 31364: 31362: 31361: 31360: 31341: 31340: 31274: 31259: 31235: 31230: 31211: 31120: 31093:10.2307/2316199 31065:10.2307/2323391 31020: 31002:Springer-Verlag 30988: 30986:Further reading 30983: 30691: 30686: 30680: 30651: 30624: 30604: 30598: 30578: 30572: 30545: 30535:10.1007/b138372 30516: 30495: 30475: 30449: 30429: 30399: 30380: 30353: 30319: 30298: 30277: 30241: 30219: 30200: 30175: 30170: 30169: 30132: 30128: 30099: 30095: 30058: 30054: 30039:10.2307/1969750 30023: 30019: 30004:10.2307/2372375 29988: 29984: 29969:10.2307/1969701 29953: 29949: 29912: 29908: 29901: 29879: 29875: 29868: 29846: 29842: 29807: 29803: 29796: 29774: 29770: 29743: 29739: 29694: 29690: 29627: 29623: 29608:10.2307/1969802 29592: 29588: 29573:10.2307/2372120 29557: 29553: 29522: 29504:Springer-Verlag 29491: 29487: 29480: 29456: 29452: 29445: 29429: 29425: 29418: 29408:10.1090/gsm/005 29394: 29390: 29375:10.2307/2007052 29359: 29355: 29326: 29322: 29302: 29298: 29259: 29255: 29226: 29222: 29195: 29191: 29176:10.2307/2946547 29160: 29156: 29141:10.2307/2372949 29125: 29121: 29084: 29080: 29073: 29049: 29045: 29010: 29006: 28989: 28978: 28947: 28943: 28912: 28908: 28886:10.2307/1970257 28870: 28859: 28828: 28824: 28783: 28776: 28753: 28749: 28720: 28716: 28687: 28683: 28648: 28644: 28610: 28604: 28600: 28562: 28558: 28517: 28513: 28498: 28470:Springer Nature 28462: 28458: 28427: 28423: 28385: 28376: 28362: 28339: 28332: 28313: 28309: 28280: 28276: 28247: 28243: 28236: 28218: 28214: 28185: 28176: 28143: 28139: 28132: 28110: 28106: 28098: 28092: 28088: 28081: 28065: 28061: 28054: 28038: 28034: 27993: 27987: 27983: 27976: 27952: 27948: 27941: 27917: 27913: 27870: 27866: 27844:10.2307/1969915 27827: 27818: 27809: 27794: 27770: 27766: 27751: 27727: 27723: 27682: 27675: 27636: 27631: 27630: 27628: 27625: 27624: 27619: 27615: 27592: 27588: 27550: 27546: 27532:10.2307/1969189 27516: 27512: 27472: 27468: 27448: 27444: 27402: 27398: 27369: 27365: 27326: 27322: 27287: 27283: 27260: 27253: 27213: 27211: 27208: 27207: 27202: 27198: 27178: 27174: 27145: 27138: 27106: 27095: 27066: 27062: 27033: 27029: 27022: 27006: 27002: 26995: 26979: 26975: 26968: 26957: 26949:10.1090/noti798 26928: 26924: 26895: 26891: 26860: 26856: 26849: 26823: 26816: 26801: 26794: 26771: 26767: 26760: 26738: 26737: 26733: 26694: 26681: 26666: 26662: 26635: 26628: 26599: 26588: 26543: 26536: 26524: 26492: 26483: 26472: 26468: 26460: 26454: 26450: 26409: 26398: 26366: 26362: 26330: 26321: 26285: 26278: 26249: 26245: 26208: 26204: 26167: 26163: 26138: 26134: 26105: 26101: 26054: 26050: 26011: 26007: 25990: 25983: 25954: 25947: 25916: 25912: 25905: 25881: 25877: 25840: 25836: 25829: 25805: 25801: 25794: 25778:. p. 494. 25766: 25762: 25725: 25721: 25714: 25692: 25685: 25662: 25658: 25638: 25634: 25605: 25601: 25543: 25534: 25527: 25505: 25501: 25470: 25461: 25408: 25395: 25358: 25337: 25300: 25293: 25286: 25270: 25266: 25212: 25211: 25209: 25206: 25205: 25200: 25193: 25188: 25183: 25178: 25177: 25158: 25156: 25153: 25152: 25135: 25130: 25129: 25127: 25124: 25123: 25121: 25117: 25111: 25107: 25094: 25090: 25070: 25065: 25064: 25062: 25059: 25058: 25041: 25036: 25035: 25026: 25021: 25020: 25018: 25015: 25014: 25013: 25009: 25004: 25000: 24995: 24991: 24986: 24982: 24977: 24973: 24968: 24964: 24959: 24955: 24947: 24943: 24938: 24934: 24928: 24924: 24919: 24915: 24910: 24906: 24882: 24877: 24876: 24874: 24871: 24870: 24868: 24864: 24851: 24847: 24842: 24838: 24833: 24829: 24820: 24816: 24810: 24806: 24786: 24782: 24780: 24777: 24776: 24770: 24766: 24757: 24753: 24748: 24744: 24731: 24727: 24712: 24708: 24703: 24656: 24634: 24626: 24625: 24623: 24620: 24619: 24578:Kähler manifold 24544: 24536: 24535: 24533: 24530: 24529: 24476: 24468: 24467: 24453: 24451: 24450: 24448: 24445: 24444: 24428: 24426: 24423: 24422: 24419:algebraic curve 24407: 24364: 24318: 24314: 24312: 24309: 24308: 24267: 24263: 24261: 24258: 24257: 24254: 24217: 24213: 24204: 24200: 24198: 24195: 24194: 24123: 24120: 24119: 24095: 24092: 24091: 24074:are compact in 24026: 24023: 24022: 23991: 23990: 23982: 23979: 23978: 23954: 23951: 23950: 23902: 23899: 23898: 23880:Riemann surface 23823: 23818: 23817: 23815: 23812: 23811: 23776: 23771: 23770: 23768: 23765: 23764: 23744: 23739: 23738: 23736: 23733: 23732: 23728: 23670: 23666: 23651: 23647: 23624: 23621: 23620: 23603: 23599: 23597: 23594: 23593: 23573: 23569: 23555: 23552: 23551: 23534: 23529: 23528: 23526: 23523: 23522: 23501: 23497: 23482: 23478: 23476: 23473: 23472: 23449: 23445: 23443: 23440: 23439: 23425: 23395: 23391: 23384: 23377: 23365: 23362: 23361: 23313: 23310: 23309: 23267: 23264: 23263: 23238: 23235: 23234: 23209: 23206: 23205: 23183: 23180: 23179: 23113: 23110: 23109: 23077: 23072: 23071: 23069: 23066: 23065: 23019: 23016: 23015: 23005: 22968: 22953: 22949: 22947: 22944: 22943: 22917: 22911: 22910: 22909: 22894: 22890: 22878: 22872: 22871: 22870: 22855: 22851: 22849: 22846: 22845: 22818: 22812: 22811: 22810: 22795: 22791: 22780: 22765: 22761: 22749: 22744: 22738: 22737: 22721: 22717: 22705: 22699: 22698: 22697: 22682: 22678: 22676: 22673: 22672: 22639: 22634: 22628: 22627: 22611: 22607: 22605: 22602: 22601: 22571:Riemann surface 22563: 22528: 22527: 22525: 22522: 22521: 22472: 22469: 22468: 22442: 22441: 22433: 22430: 22429: 22403: 22400: 22399: 22345: 22344: 22324: 22307: 22295: 22283: 22266: 22253: 22249: 22235: 22234: 22232: 22229: 22228: 22200: 22197: 22196: 22151: 22150: 22148: 22145: 22144: 22128: 22125: 22124: 22110: 22079: 22074: 22073: 22071: 22068: 22067: 22019: 22014: 22013: 22011: 22008: 22007: 21982: 21977: 21976: 21974: 21971: 21970: 21953: 21948: 21947: 21945: 21942: 21941: 21916: 21911: 21910: 21908: 21905: 21904: 21884: 21879: 21878: 21876: 21873: 21872: 21840: 21838: 21835: 21834: 21823: 21818: 21788: 21773: 21769: 21767: 21764: 21763: 21741: 21738: 21737: 21712: 21697: 21693: 21691: 21688: 21687: 21658: 21643: 21639: 21628: 21613: 21609: 21588: 21583: 21582: 21567: 21563: 21552: 21537: 21533: 21531: 21528: 21527: 21505: 21490: 21486: 21484: 21481: 21480: 21460: 21449: 21446: 21445: 21419: 21414: 21413: 21398: 21390: 21373: 21370: 21369: 21347: 21332: 21328: 21326: 21323: 21322: 21305: 21300: 21299: 21297: 21294: 21293: 21273: 21268: 21267: 21252: 21248: 21246: 21243: 21242: 21214: 21209: 21208: 21193: 21189: 21187: 21184: 21183: 21160: 21155: 21154: 21139: 21135: 21123: 21118: 21117: 21112: 21106: 21101: 21100: 21085: 21081: 21069: 21060: 21055: 21054: 21039: 21035: 21033: 21030: 21029: 21000: 20995: 20994: 20989: 20983: 20978: 20977: 20962: 20958: 20946: 20937: 20932: 20931: 20916: 20912: 20910: 20907: 20906: 20887: 20884: 20883: 20866: 20861: 20860: 20855: 20849: 20844: 20843: 20841: 20838: 20837: 20816: 20811: 20810: 20808: 20805: 20804: 20787: 20782: 20781: 20779: 20776: 20775: 20755: 20751: 20746: 20741: 20738: 20737: 20717: 20714: 20713: 20696: 20692: 20687: 20681: 20677: 20675: 20672: 20671: 20668: 20621: 20606: 20602: 20591: 20586: 20581: 20566: 20562: 20550: 20542: 20527: 20523: 20521: 20518: 20517: 20485: 20480: 20475: 20460: 20456: 20444: 20436: 20421: 20417: 20415: 20412: 20411: 20373: 20370: 20369: 20352: 20348: 20346: 20343: 20342: 20321: 20317: 20309: 20306: 20305: 20285: 20282: 20281: 20264: 20260: 20251: 20247: 20245: 20242: 20241: 20224: 20220: 20218: 20215: 20214: 20197: 20193: 20191: 20188: 20187: 20180: 20143: 20117: 20111: 20110: 20109: 20107: 20104: 20103: 20095:. (Cartan 1950) 20079: 20074: 20073: 20071: 20068: 20067: 20041: 20040: 20038: 20035: 20034: 20012: 20006: 20005: 20004: 19993: 19990: 19989: 19972: 19966: 19965: 19964: 19962: 19959: 19958: 19941: 19936: 19935: 19933: 19930: 19929: 19906: 19901: 19900: 19899: 19893: 19892: 19891: 19882: 19881: 19879: 19876: 19875: 19868: 19845: 19839: 19838: 19837: 19835: 19832: 19831: 19814: 19813: 19811: 19808: 19807: 19784: 19779: 19778: 19772: 19766: 19765: 19764: 19755: 19750: 19749: 19743: 19737: 19736: 19735: 19726: 19721: 19720: 19714: 19708: 19707: 19706: 19698: 19695: 19694: 19666: 19660: 19659: 19658: 19656: 19653: 19652: 19632: 19629: 19628: 19611: 19605: 19604: 19603: 19601: 19598: 19597: 19580: 19575: 19574: 19568: 19567: 19558: 19553: 19552: 19543: 19538: 19532: 19531: 19522: 19519: 19518: 19496: 19493: 19492: 19470: 19467: 19466: 19447: 19444: 19443: 19426: 19421: 19420: 19414: 19413: 19404: 19399: 19398: 19389: 19384: 19378: 19377: 19374: 19371: 19370: 19354: 19351: 19350: 19334: 19331: 19330: 19311: 19308: 19307: 19290: 19284: 19283: 19282: 19280: 19277: 19276: 19255: 19254: 19252: 19249: 19248: 19228: 19227: 19225: 19222: 19221: 19201: 19195: 19194: 19193: 19182: 19179: 19178: 19155: 19152: 19151: 19135: 19132: 19131: 19105: 19100: 19099: 19093: 19092: 19083: 19078: 19077: 19068: 19063: 19057: 19056: 19046: 19041: 19040: 19031: 19026: 19020: 19019: 19016: 19013: 19012: 18990: 18987: 18986: 18970: 18967: 18966: 18945: 18939: 18938: 18937: 18935: 18932: 18931: 18914: 18913: 18911: 18908: 18907: 18887: 18881: 18880: 18879: 18868: 18865: 18864: 18854: 18849: 18800: 18797: 18796: 18756: 18739: 18728: 18725: 18724: 18690: 18679: 18656: 18653: 18652: 18615: 18595: 18592: 18591: 18569: 18558: 18555: 18554: 18514: 18511: 18510: 18490: 18487: 18486: 18470: 18467: 18466: 18438: 18435: 18434: 18412: 18409: 18408: 18405: 18393: 18362: 18361: 18359: 18356: 18355: 18334: 18329: 18328: 18326: 18323: 18322: 18292: 18289: 18288: 18266: 18263: 18262: 18240: 18237: 18236: 18208: 18205: 18204: 18144: 18139: 18138: 18130: 18127: 18126: 18123: 18102: 18099: 18098: 18082: 18079: 18078: 18061: 18056: 18055: 18053: 18050: 18049: 18024: 18018: 18017: 18016: 18014: 18011: 18010: 17981: 17977: 17975: 17972: 17971: 17949: 17946: 17945: 17917: 17914: 17913: 17883: 17880: 17879: 17842: 17839: 17838: 17835: 17813: 17808: 17807: 17805: 17802: 17801: 17762: 17759: 17758: 17755: 17715: 17712: 17711: 17662: 17659: 17658: 17633: 17625: 17601: 17597: 17588: 17584: 17564: 17561: 17560: 17531: 17523: 17499: 17495: 17486: 17482: 17462: 17459: 17458: 17457:. Then the set 17442: 17439: 17438: 17417: 17410: 17406: 17402: 17400: 17398: 17395: 17394: 17368: 17360: 17358: 17355: 17354: 17314: 17306: 17298: 17295: 17294: 17262: 17258: 17249: 17245: 17237: 17234: 17233: 17226: 17221: 17211:itself. Domain 17196: 17193: 17192: 17176: 17173: 17172: 17152: 17149: 17148: 17131: 17130: 17128: 17125: 17124: 17101: 17098: 17097: 17094: 17086: 17064: 17058: 17057: 17056: 17054: 17051: 17050: 17034: 17031: 17030: 17007: 17003: 16997: 16986: 16974: 16971: 16970: 16945: 16939: 16938: 16937: 16935: 16932: 16931: 16908: 16904: 16902: 16899: 16898: 16866: 16860: 16859: 16858: 16856: 16853: 16852: 16816: 16812: 16810: 16809: 16803: 16799: 16784: 16780: 16778: 16777: 16767: 16763: 16759: 16740: 16736: 16735: 16733: 16727: 16710: 16689: 16686: 16685: 16660: 16656: 16647: 16643: 16639: 16622: 16620: 16614: 16603: 16576: 16573: 16572: 16553: 16550: 16549: 16524: 16521: 16520: 16479: 16476: 16475: 16441: 16438: 16437: 16420: 16414: 16413: 16412: 16410: 16407: 16406: 16390: 16381: 16376: 16375: 16367: 16364: 16363: 16342: 16336: 16335: 16334: 16332: 16329: 16328: 16325: 16320: 16295: 16292: 16291: 16265: 16259: 16258: 16257: 16240: 16232: 16229: 16228: 16209: 16178: 16172: 16171: 16170: 16153: 16145: 16142: 16141: 16107: 16101: 16100: 16099: 16082: 16074: 16071: 16070: 16006: 16003: 16002: 15982: 15979: 15978: 15947: 15941: 15940: 15939: 15931: 15928: 15927: 15924: 15898: 15892: 15891: 15890: 15888: 15885: 15884: 15877: 15843: 15842: 15829: 15827: 15824: 15823: 15802: 15796: 15795: 15794: 15792: 15789: 15788: 15765: 15754: 15753: 15752: 15742: 15738: 15734: 15724: 15720: 15719: 15717: 15705: 15701: 15686: 15682: 15670: 15666: 15664: 15661: 15660: 15641: 15638: 15637: 15620: 15614: 15613: 15612: 15610: 15607: 15606: 15590: 15587: 15586: 15552: 15541: 15531: 15527: 15526: 15524: 15520: 15509: 15506: 15505: 15470: 15467: 15466: 15447: 15439: 15436: 15435: 15401: 15381: 15378: 15377: 15346: 15343: 15342: 15337:holomorphic map 15303: 15289: 15286: 15285: 15279:as follows. An 15264: 15261: 15260: 15257:full generality 15209: 15198: 15195: 15194: 15150: 15147: 15146: 15125: 15120: 15119: 15105: 15102: 15101: 15084: 15079: 15078: 15067: 15041: 15038: 15037: 15006: 15000: 14999: 14998: 14990: 14987: 14986: 14946: 14945: 14931: 14928: 14927: 14921: 14913:convex function 14880: 14876: 14868: 14857: 14854: 14853: 14833: 14829: 14815: 14809: 14805: 14800: 14798: 14795: 14794: 14771: 14767: 14765: 14762: 14761: 14744: 14739: 14738: 14736: 14733: 14732: 14716: 14713: 14712: 14688: 14686: 14683: 14682: 14671: 14669:Pseudoconvexity 14646: 14641: 14640: 14629: 14626: 14625: 14602: 14597: 14596: 14587: 14583: 14581: 14578: 14577: 14557: 14553: 14539: 14537: 14534: 14533: 14505: 14501: 14492: 14481: 14480: 14479: 14471: 14459: 14455: 14441: 14439: 14436: 14435: 14405: 14402: 14401: 14370: 14367: 14366: 14345: 14334: 14333: 14332: 14330: 14327: 14326: 14310: 14307: 14306: 14305:, every domain 14284: 14281: 14280: 14252: 14241: 14240: 14239: 14231: 14228: 14227: 14207: 14204: 14203: 14170: 14169: 14167: 14164: 14163: 14119: 14118: 14107: 14102: 14085: 14073: 14061: 14044: 14031: 14027: 14018: 14007: 14006: 14005: 14003: 14000: 13999: 13966: 13963: 13962: 13932: 13931: 13929: 13926: 13925: 13905: 13902: 13901: 13885: 13882: 13881: 13864: 13859: 13858: 13850: 13847: 13846: 13843: 13815: 13812: 13811: 13787: 13783: 13774: 13770: 13768: 13765: 13764: 13747: 13743: 13741: 13738: 13737: 13720: 13716: 13714: 13711: 13710: 13686: 13682: 13676: 13665: 13653: 13650: 13649: 13626: 13622: 13613: 13609: 13607: 13604: 13603: 13583: 13579: 13573: 13562: 13550: 13547: 13546: 13529: 13525: 13510: 13506: 13504: 13501: 13500: 13496: 13463: 13455: 13452: 13451: 13429: 13426: 13425: 13396: 13393: 13392: 13348: 13345: 13344: 13322: 13319: 13318: 13301: 13296: 13295: 13287: 13284: 13283: 13261: 13258: 13257: 13236: 13231: 13230: 13228: 13225: 13224: 13178: 13173: 13172: 13164: 13161: 13160: 13117: 13114: 13113: 13092: 13087: 13086: 13078: 13075: 13074: 13047: 13044: 13043: 13036: 12996: 12991: 12990: 12988: 12985: 12984: 12967: 12962: 12961: 12959: 12956: 12955: 12934: 12929: 12928: 12926: 12923: 12922: 12918: 12895: 12891: 12879: 12875: 12867: 12864: 12863: 12847: 12844: 12843: 12817: 12813: 12795: 12791: 12785: 12781: 12772: 12768: 12766: 12763: 12762: 12745: 12740: 12739: 12730: 12725: 12724: 12716: 12713: 12712: 12695: 12691: 12689: 12686: 12685: 12668: 12664: 12662: 12659: 12658: 12644: 12582: 12577: 12576: 12568: 12559: 12554: 12553: 12545: 12533: 12528: 12527: 12504: 12501: 12500: 12472: 12467: 12466: 12458: 12449: 12444: 12443: 12435: 12423: 12418: 12417: 12394: 12391: 12390: 12363: 12355: 12338: 12330: 12318: 12313: 12312: 12289: 12286: 12285: 12272: 12250: 12246: 12244: 12241: 12240: 12223: 12219: 12217: 12214: 12213: 12196: 12192: 12190: 12187: 12186: 12169: 12165: 12163: 12160: 12159: 12142: 12138: 12136: 12133: 12132: 12115: 12111: 12109: 12106: 12105: 12073: 12055: 12038: 12033: 12032: 12030: 12027: 12026: 12022: 12012: 11952: 11946: 11942: 11937: 11905: 11899: 11895: 11890: 11881: 11877: 11865: 11861: 11852: 11848: 11827: 11823: 11821: 11818: 11817: 11784: 11781: 11780: 11755: 11749: 11745: 11740: 11726: 11720: 11716: 11711: 11702: 11697: 11696: 11678: 11674: 11672: 11669: 11668: 11647: 11642: 11641: 11639: 11636: 11635: 11632: 11627: 11600: 11597: 11596: 11579: 11575: 11560: 11556: 11554: 11551: 11550: 11528: 11524: 11523: 11519: 11513: 11509: 11500: 11496: 11482: 11478: 11477: 11473: 11467: 11463: 11454: 11450: 11439: 11435: 11420: 11416: 11415: 11411: 11405: 11392: 11388: 11373: 11369: 11368: 11362: 11359: 11358: 11341: 11336: 11335: 11333: 11330: 11329: 11309: 11304: 11303: 11301: 11298: 11297: 11271: 11265: 11261: 11256: 11236: 11230: 11226: 11221: 11183: 11180: 11179: 11147: 11143: 11128: 11124: 11106: 11102: 11087: 11083: 11062: 11058: 11056: 11053: 11052: 11029: 11025: 11017: 11014: 11013: 10999: 10910: 10906: 10897: 10892: 10887: 10883: 10869: 10865: 10856: 10852: 10851: 10847: 10835: 10831: 10816: 10812: 10802: 10798: 10796: 10793: 10792: 10766: 10762: 10760: 10757: 10756: 10693: 10689: 10672: 10668: 10667: 10663: 10639: 10635: 10620: 10616: 10606: 10602: 10600: 10597: 10596: 10570: 10566: 10564: 10561: 10560: 10532: 10528: 10519: 10514: 10499: 10495: 10491: 10487: 10486: 10482: 10468: 10464: 10455: 10451: 10450: 10446: 10444: 10441: 10440: 10391: 10387: 10385: 10382: 10381: 10314: 10310: 10301: 10296: 10291: 10287: 10273: 10269: 10260: 10256: 10255: 10251: 10239: 10235: 10220: 10216: 10206: 10202: 10200: 10197: 10196: 10167: 10162: 10143: 10138: 10122: 10118: 10116: 10113: 10112: 10095: 10090: 10089: 10077: 10073: 10058: 10054: 10043: 10040: 10039: 10017: 10014: 10013: 9996: 9991: 9990: 9982: 9979: 9978: 9971: 9932: 9929: 9928: 9908: 9905: 9904: 9878: 9877: 9869: 9866: 9865: 9839: 9838: 9830: 9827: 9826: 9806: 9803: 9802: 9775: 9772: 9771: 9730: 9725: 9724: 9712: 9707: 9706: 9701: 9698: 9697: 9675: 9672: 9671: 9651: 9648: 9647: 9601: 9598: 9597: 9596:.) assume that 9567: 9566: 9564: 9561: 9560: 9534: 9533: 9525: 9522: 9521: 9495: 9494: 9486: 9483: 9482: 9465: 9460: 9459: 9457: 9454: 9453: 9446: 9399: 9396: 9395: 9392: 9355: 9352: 9351: 9319: 9315: 9313: 9310: 9309: 9281: 9278: 9277: 9260: 9255: 9254: 9252: 9249: 9248: 9235:For the domain 9230: 9187: 9183: 9181: 9178: 9177: 9160: 9155: 9154: 9152: 9149: 9148: 9131: 9127: 9115: 9111: 9102: 9098: 9090: 9087: 9086: 9060: 9056: 9047: 9043: 9035: 9032: 9031: 9002: 8997: 8996: 8988: 8985: 8984: 8977: 8918: 8914: 8896: 8893: 8892: 8824: 8818: 8814: 8772: 8768: 8750: 8747: 8746: 8728: 8724: 8712: 8702: 8701: 8683: 8679: 8661: 8657: 8645: 8635: 8634: 8621: 8611: 8610: 8601: 8591: 8590: 8569: 8553: 8548: 8547: 8533: 8532: 8527: 8518: 8514: 8501: 8481: 8479: 8456: 8453: 8452: 8415: 8412: 8411: 8395: 8392: 8391: 8356: 8353: 8352: 8335: 8330: 8329: 8327: 8324: 8323: 8303: 8300: 8299: 8271: 8268: 8267: 8248: 8245: 8244: 8227: 8222: 8221: 8219: 8216: 8215: 8197: 8175: 8171: 8159: 8153: 8150: 8149: 8132: 8128: 8116: 8110: 8107: 8106: 8086: 8074: 8066: 8054: 8048: 8045: 8044: 8027: 8026: 8011: 8007: 7992: 7988: 7974: 7970: 7965: 7956: 7952: 7921: 7917: 7916: 7911: 7890: 7886: 7885: 7880: 7860: 7856: 7844: 7840: 7839: 7831: 7828: 7815: 7811: 7782: 7778: 7770: 7764: 7760: 7755: 7754: 7750: 7737: 7732: 7722: 7717: 7711: 7700: 7687: 7686: 7680: 7676: 7667: 7663: 7648: 7630: 7625: 7616: 7612: 7608: 7602: 7598: 7596: 7595: 7591: 7590: 7561: 7557: 7549: 7543: 7539: 7534: 7533: 7529: 7520: 7516: 7503: 7498: 7488: 7483: 7477: 7466: 7455: 7439: 7437: 7434: 7433: 7388: 7379: 7375: 7367: 7359: 7350: 7346: 7337: 7332: 7331: 7319: 7315: 7300: 7296: 7287: 7283: 7273: 7269: 7267: 7264: 7263: 7235: 7232: 7231: 7228: 7177: 7168: 7164: 7156: 7150: 7146: 7137: 7133: 7128: 7119: 7114: 7113: 7101: 7097: 7082: 7078: 7069: 7065: 7055: 7051: 7049: 7046: 7045: 7003: 6994: 6990: 6982: 6976: 6972: 6963: 6959: 6954: 6945: 6940: 6939: 6927: 6923: 6908: 6904: 6895: 6891: 6881: 6877: 6875: 6872: 6871: 6849: 6845: 6844: 6840: 6834: 6830: 6821: 6817: 6803: 6799: 6798: 6794: 6788: 6784: 6775: 6771: 6760: 6756: 6741: 6737: 6736: 6732: 6726: 6713: 6709: 6694: 6690: 6689: 6683: 6680: 6679: 6629: 6625: 6620: 6617: 6616: 6613: 6586: 6582: 6581: 6574: 6570: 6569: 6568: 6554: 6550: 6549: 6542: 6538: 6537: 6536: 6532: 6525: 6521: 6513: 6509: 6494: 6490: 6489: 6485: 6484: 6482: 6476: 6465: 6447: 6443: 6442: 6435: 6431: 6430: 6429: 6415: 6411: 6410: 6403: 6399: 6398: 6397: 6393: 6381: 6377: 6362: 6358: 6357: 6353: 6352: 6350: 6348: 6345: 6344: 6315: 6311: 6309: 6306: 6305: 6284: 6280: 6278: 6275: 6274: 6249: 6245: 6230: 6226: 6224: 6221: 6220: 6187: 6183: 6182: 6178: 6172: 6168: 6159: 6155: 6141: 6137: 6136: 6132: 6126: 6122: 6113: 6109: 6098: 6094: 6079: 6075: 6074: 6070: 6064: 6051: 6047: 6032: 6028: 6027: 6006: 6003: 6002: 5970: 5967: 5966: 5949: 5944: 5943: 5925: 5921: 5906: 5902: 5891: 5888: 5887: 5886:For each point 5863: 5862: 5856: 5852: 5840: 5836: 5815: 5811: 5810: 5806: 5800: 5796: 5787: 5783: 5763: 5759: 5758: 5754: 5748: 5744: 5735: 5731: 5727: 5717: 5713: 5698: 5694: 5687: 5685: 5677: 5673: 5669: 5665: 5654: 5650: 5646: 5642: 5633: 5629: 5616: 5611: 5600: 5596: 5587: 5583: 5582: 5578: 5574: 5573: 5559: 5555: 5554: 5550: 5544: 5540: 5531: 5527: 5513: 5509: 5508: 5504: 5498: 5494: 5485: 5481: 5470: 5466: 5451: 5447: 5446: 5442: 5436: 5423: 5419: 5404: 5400: 5399: 5375: 5373: 5370: 5369: 5326: 5317: 5313: 5305: 5299: 5295: 5286: 5282: 5277: 5268: 5263: 5262: 5250: 5246: 5231: 5227: 5218: 5214: 5200: 5197: 5196: 5189: 5155: 5151: 5150: 5143: 5139: 5138: 5137: 5126: 5122: 5121: 5114: 5110: 5109: 5108: 5107: 5097: 5093: 5084: 5080: 5076: 5074: 5056: 5052: 5051: 5044: 5040: 5039: 5038: 5024: 5020: 5019: 5012: 5008: 5004: 5003: 5002: 4992: 4988: 4973: 4969: 4960: 4956: 4942: 4938: 4923: 4919: 4918: 4914: 4913: 4911: 4907: 4905: 4902: 4901: 4882: 4874: 4866: 4864: 4861: 4860: 4818: 4809: 4805: 4797: 4791: 4787: 4778: 4774: 4769: 4760: 4755: 4754: 4742: 4738: 4723: 4719: 4710: 4706: 4696: 4692: 4690: 4687: 4686: 4662: 4656: 4655: 4654: 4652: 4649: 4648: 4609: 4605: 4593: 4589: 4568: 4564: 4563: 4559: 4553: 4549: 4540: 4536: 4516: 4512: 4511: 4507: 4501: 4497: 4488: 4484: 4480: 4470: 4466: 4451: 4447: 4440: 4438: 4430: 4426: 4422: 4418: 4407: 4403: 4399: 4395: 4386: 4382: 4369: 4359: 4355: 4346: 4342: 4341: 4339: 4325: 4321: 4320: 4313: 4309: 4308: 4307: 4293: 4289: 4288: 4281: 4277: 4276: 4275: 4271: 4261: 4257: 4242: 4238: 4229: 4225: 4211: 4207: 4192: 4188: 4187: 4183: 4182: 4180: 4178: 4175: 4174: 4160: 4125: 4121: 4109: 4105: 4089: 4085: 4076: 4072: 4057: 4053: 4044: 4040: 4036: 4026: 4022: 4007: 4003: 3996: 3994: 3986: 3982: 3978: 3974: 3963: 3959: 3955: 3951: 3942: 3938: 3925: 3920: 3908: 3904: 3889: 3885: 3877: 3874: 3873: 3837: 3834: 3833: 3816: 3815: 3809: 3805: 3789: 3785: 3776: 3772: 3757: 3753: 3744: 3740: 3728: 3724: 3715: 3711: 3707: 3697: 3693: 3678: 3674: 3665: 3661: 3654: 3652: 3644: 3640: 3636: 3632: 3626: 3622: 3610: 3606: 3602: 3598: 3589: 3585: 3573: 3569: 3565: 3561: 3552: 3548: 3535: 3530: 3521: 3520: 3514: 3510: 3494: 3490: 3481: 3477: 3465: 3461: 3452: 3448: 3444: 3434: 3430: 3415: 3411: 3402: 3398: 3389: 3385: 3378: 3376: 3368: 3364: 3360: 3356: 3350: 3346: 3334: 3330: 3326: 3322: 3313: 3309: 3296: 3291: 3282: 3281: 3275: 3271: 3258: 3254: 3245: 3241: 3240: 3230: 3226: 3211: 3207: 3198: 3194: 3187: 3185: 3177: 3173: 3169: 3165: 3152: 3147: 3140: 3131: 3127: 3112: 3108: 3098: 3096: 3093: 3092: 3068: 3057: 3045: 3042: 3041: 2999: 2993: 2989: 2975: 2971: 2962: 2958: 2957: 2953: 2944: 2939: 2938: 2926: 2922: 2907: 2903: 2894: 2890: 2880: 2876: 2848: 2846: 2843: 2842: 2824: 2820: 2805: 2801: 2792: 2788: 2787: 2774: 2772: 2769: 2768: 2747: 2745: 2742: 2741: 2709: 2705: 2690: 2686: 2677: 2673: 2672: 2670: 2668: 2665: 2664: 2641: 2637: 2622: 2618: 2609: 2605: 2604: 2602: 2600: 2597: 2596: 2579: 2575: 2573: 2570: 2569: 2552: 2548: 2546: 2543: 2542: 2502: 2499: 2498: 2481: 2475: 2474: 2473: 2471: 2468: 2467: 2446: 2442: 2440: 2437: 2436: 2420: 2417: 2416: 2402: 2367: 2366: 2364: 2361: 2360: 2322: 2318: 2316: 2312: 2304: 2302: 2266: 2263: 2262: 2236: 2232: 2228: 2220: 2218: 2203: 2199: 2195: 2187: 2185: 2181: and 2179: 2169: 2165: 2161: 2153: 2151: 2139: 2135: 2131: 2123: 2121: 2085: 2082: 2081: 2065: 2062: 2061: 2045: 2042: 2041: 2025: 2022: 2021: 2005: 1996: 1991: 1990: 1982: 1979: 1978: 1946: 1938: 1936: 1911: 1903: 1901: 1897: and 1895: 1875: 1867: 1865: 1843: 1835: 1833: 1831: 1828: 1827: 1811: 1808: 1807: 1787: 1784: 1783: 1767: 1764: 1763: 1747: 1739: 1736: 1735: 1719: 1711: 1703: 1700: 1699: 1696: 1641: 1637: 1616: 1612: 1591: 1587: 1572: 1568: 1554: 1551: 1550: 1449: 1446: 1445: 1427: 1418: 1413: 1412: 1404: 1401: 1400: 1378: 1375: 1374: 1358: 1356: 1353: 1352: 1335: 1330: 1329: 1321: 1318: 1317: 1310: 1305: 1283: 1278: 1277: 1275: 1272: 1271: 1254: 1249: 1248: 1246: 1243: 1242: 1199: 1194: 1193: 1185: 1176: 1172: 1163: 1159: 1157: 1154: 1153: 1123: 1122: 1117: 1111: 1110: 1102: 1092: 1091: 1089: 1086: 1085: 1065: 1058:Cartesian plane 1042: 1022: 1018: 1016:linear operator 1001: 977: 972: 971: 969: 966: 965: 945: 940: 939: 937: 934: 933: 913: 911: 908: 907: 900: 897:complex numbers 890: 886:Kähler manifold 864: 859: 858: 856: 853: 852: 831: 826: 825: 823: 820: 819: 802: 797: 796: 794: 791: 790: 774: 772: 769: 768: 762: 741: 736: 735: 733: 730: 729: 723: 684: 572:Stein manifolds 566:pseudoconvexity 547: 542: 541: 539: 536: 535: 524: 508: 506: 503: 502: 472:double integral 460: 449: 433: 424: 419: 418: 410: 407: 406: 379:Heinrich Behnke 362: 349:Riemann surface 313:theta functions 305: 293:Stein manifolds 275: 270: 269: 267: 264: 263: 246: 238: 237: 235: 232: 231: 191: 186: 185: 177: 174: 173: 153: 145: 142: 141: 129: 114: 110: 89: 67:complex numbers 62: 45: 40: 39: 37: 34: 33: 17: 12: 11: 5: 31369: 31359: 31358: 31353: 31339: 31338: 31293: 31278: 31272: 31252: 31242: 31234: 31233:External links 31231: 31229: 31228: 31204: 31192:(2): 117–142. 31177: 31165:(2): 101–118. 31150: 31123: 31118: 31105: 31087:(7): 681–703. 31076: 31059:(3): 236–256, 31048: 31018: 30989: 30987: 30984: 30982: 30981: 30963: 30945: 30927: 30914:"Oka theorems" 30909: 30891: 30873: 30855: 30837: 30819: 30801: 30783: 30765: 30747: 30729: 30716:"Power series" 30711: 30692: 30690: 30687: 30685: 30684: 30678: 30663: 30649: 30628: 30622: 30602: 30596: 30576: 30570: 30549: 30543: 30520: 30514: 30499: 30493: 30478: 30473: 30459:Lars Hörmander 30455: 30454: 30453: 30447: 30410: 30397: 30384: 30378: 30357: 30351: 30325: 30324: 30323: 30317: 30296: 30283: 30282: 30281: 30275: 30252:(in Russian). 30245: 30239: 30223: 30217: 30204: 30198: 30176: 30174: 30171: 30168: 30167: 30126: 30113:(1): 107–121. 30093: 30052: 30033:(3): 494–500. 30017: 29998:(2): 893–914. 29982: 29947: 29906: 29899: 29873: 29866: 29840: 29801: 29794: 29768: 29737: 29688: 29641:(6): 522–527. 29621: 29602:(2): 298–342. 29586: 29567:(4): 813–875. 29551: 29520: 29485: 29478: 29450: 29443: 29423: 29416: 29388: 29369:(2): 229–244. 29353: 29320: 29306:"Levi problem" 29296: 29275:(4): 876–880. 29253: 29240:(5): 712–716. 29220: 29189: 29170:(1): 123–135. 29154: 29135:(4): 917–934. 29119: 29098:(2): 345–366. 29078: 29071: 29043: 29024:(2): 529–547. 29004: 28976: 28957:(3): 195–216. 28941: 28922:(4): 355–365. 28906: 28880:(2): 460–472, 28857: 28838:(5): 377–391. 28822: 28774: 28747: 28714: 28681: 28642: 28598: 28556: 28527:(1–2): 43–45. 28511: 28496: 28456: 28437:(2): 103–108. 28421: 28374: 28360: 28330: 28315:Serre, J. -P. 28307: 28274: 28241: 28234: 28212: 28174: 28137: 28130: 28104: 28086: 28079: 28059: 28052: 28032: 28004:(3): 566–586, 27998:Kodai Math. J. 27981: 27974: 27946: 27939: 27911: 27864: 27838:(2): 197–278, 27807: 27792: 27764: 27749: 27721: 27673: 27662:(3): 275–277. 27639: 27634: 27613: 27586: 27544: 27526:(3): 472–542, 27510: 27466: 27452:"Modification" 27442: 27396: 27363: 27320: 27281: 27264:RIMS Kôkyûroku 27251: 27216: 27196: 27172: 27136: 27116:(2): 177–195, 27093: 27060: 27027: 27020: 27000: 26993: 26973: 26955: 26922: 26889: 26854: 26847: 26814: 26792: 26765: 26758: 26731: 26679: 26660: 26626: 26586: 26574:(2): 259–278, 26553:(1): 204–214, 26534: 26481: 26466: 26448: 26396: 26360: 26319: 26276: 26243: 26222:(2): 111–128. 26202: 26161: 26132: 26119:(2): 574–578. 26099: 26068:(3): 513–546. 26048: 26005: 25981: 25957:Hartogs, Fritz 25945: 25926:(4): 343–353. 25910: 25903: 25875: 25834: 25827: 25799: 25792: 25760: 25719: 25712: 25683: 25656: 25632: 25599: 25569:10.5802/aif.59 25532: 25525: 25499: 25459: 25422:(4): 531–568. 25393: 25372:(4): 481–513. 25335: 25314:(3): 433–480. 25291: 25284: 25264: 25219: 25216: 25190: 25189: 25187: 25184: 25182: 25179: 25176: 25175: 25161: 25138: 25133: 25115: 25105: 25088: 25073: 25068: 25044: 25039: 25034: 25029: 25024: 25007: 24998: 24989: 24980: 24971: 24962: 24953: 24941: 24932: 24922: 24913: 24904: 24885: 24880: 24862: 24845: 24836: 24827: 24814: 24804: 24789: 24785: 24764: 24751: 24742: 24725: 24705: 24704: 24702: 24699: 24698: 24697: 24692: 24687: 24682: 24677: 24672: 24667: 24662: 24655: 24652: 24637: 24632: 24629: 24602:GAGA principle 24597:Chow's theorem 24547: 24542: 24539: 24479: 24474: 24471: 24466: 24460: 24456: 24431: 24406: 24403: 24378:, relating to 24368: 24367: 24362: 24344: 24341: 24338: 24335: 24332: 24329: 24324: 24321: 24317: 24296: 24293: 24290: 24287: 24284: 24281: 24278: 24273: 24270: 24266: 24252: 24234: 24231: 24228: 24223: 24220: 24216: 24212: 24207: 24203: 24175: 24163: 24160: 24157: 24154: 24151: 24148: 24145: 24142: 24139: 24136: 24133: 24130: 24127: 24112:Stein manifold 24099: 24086:, named after 24063: 24060: 24057: 24054: 24051: 24048: 24045: 24042: 24039: 24036: 24033: 24030: 24010: 24007: 24004: 23998: 23995: 23989: 23986: 23975:Morse function 23958: 23939: 23936: 23912: 23909: 23906: 23891: 23884:if and only if 23876: 23857: 23856: 23835: 23832: 23829: 23826: 23821: 23800: 23793: 23779: 23774: 23761: 23747: 23742: 23727: 23724: 23711: 23708: 23705: 23702: 23699: 23696: 23693: 23690: 23687: 23684: 23678: 23673: 23669: 23665: 23662: 23657: 23654: 23650: 23646: 23643: 23640: 23637: 23634: 23631: 23628: 23606: 23602: 23581: 23576: 23572: 23568: 23565: 23562: 23559: 23537: 23532: 23504: 23500: 23496: 23493: 23490: 23485: 23481: 23460: 23457: 23452: 23448: 23424: 23421: 23417: 23416: 23398: 23394: 23387: 23383: 23380: 23376: 23372: 23369: 23350: 23349: 23329: 23326: 23323: 23320: 23317: 23294: 23293: 23277: 23274: 23271: 23251: 23248: 23245: 23242: 23222: 23219: 23216: 23213: 23193: 23190: 23187: 23168: 23167: 23156: 23155: 23123: 23120: 23117: 23080: 23075: 23062: 23061: 23029: 23026: 23023: 23004: 23001: 22981: 22978: 22975: 22971: 22967: 22964: 22961: 22956: 22952: 22931: 22928: 22925: 22920: 22914: 22908: 22905: 22902: 22897: 22893: 22889: 22886: 22881: 22875: 22869: 22866: 22863: 22858: 22854: 22838: 22837: 22826: 22821: 22815: 22809: 22806: 22803: 22798: 22794: 22790: 22787: 22783: 22779: 22776: 22773: 22768: 22764: 22760: 22757: 22752: 22747: 22741: 22735: 22732: 22729: 22724: 22720: 22716: 22713: 22708: 22702: 22696: 22693: 22690: 22685: 22681: 22653: 22650: 22647: 22642: 22637: 22631: 22625: 22622: 22619: 22614: 22610: 22562: 22559: 22555: 22554: 22542: 22539: 22536: 22531: 22514: 22503: 22500: 22497: 22494: 22491: 22488: 22485: 22482: 22479: 22476: 22456: 22453: 22450: 22445: 22440: 22437: 22413: 22410: 22407: 22389: 22379: 22378: 22367: 22363: 22359: 22356: 22353: 22348: 22343: 22340: 22337: 22331: 22327: 22323: 22320: 22317: 22314: 22310: 22304: 22301: 22298: 22294: 22290: 22286: 22282: 22279: 22276: 22273: 22269: 22265: 22262: 22259: 22256: 22252: 22248: 22242: 22239: 22210: 22207: 22204: 22186:Stein manifold 22165: 22162: 22159: 22154: 22132: 22109: 22106: 22082: 22077: 22064:affine schemes 22040:Stein manifold 22022: 22017: 21985: 21980: 21956: 21951: 21919: 21914: 21890: 21887: 21882: 21843: 21822: 21819: 21817: 21814: 21801: 21798: 21795: 21791: 21787: 21784: 21781: 21776: 21772: 21751: 21748: 21745: 21725: 21722: 21719: 21715: 21711: 21708: 21705: 21700: 21696: 21680: 21679: 21668: 21665: 21661: 21657: 21654: 21651: 21646: 21642: 21638: 21635: 21631: 21627: 21624: 21621: 21616: 21612: 21608: 21605: 21602: 21599: 21596: 21591: 21586: 21581: 21578: 21575: 21570: 21566: 21562: 21559: 21555: 21551: 21548: 21545: 21540: 21536: 21512: 21508: 21504: 21501: 21498: 21493: 21489: 21463: 21459: 21456: 21453: 21442: 21441: 21430: 21427: 21422: 21417: 21410: 21406: 21401: 21397: 21393: 21389: 21386: 21383: 21380: 21377: 21354: 21350: 21346: 21343: 21340: 21335: 21331: 21308: 21303: 21281: 21276: 21271: 21266: 21263: 21260: 21255: 21251: 21228: 21225: 21222: 21217: 21212: 21207: 21204: 21201: 21196: 21192: 21180: 21179: 21168: 21163: 21158: 21153: 21150: 21147: 21142: 21138: 21134: 21131: 21126: 21121: 21115: 21109: 21104: 21099: 21096: 21093: 21088: 21084: 21077: 21073: 21068: 21063: 21058: 21053: 21050: 21047: 21042: 21038: 21023: 21022: 21011: 21008: 21003: 20998: 20992: 20986: 20981: 20976: 20973: 20970: 20965: 20961: 20954: 20950: 20945: 20940: 20935: 20930: 20927: 20924: 20919: 20915: 20891: 20869: 20864: 20858: 20852: 20847: 20834:abelian groups 20819: 20814: 20790: 20785: 20758: 20754: 20749: 20745: 20721: 20699: 20695: 20690: 20684: 20680: 20667: 20664: 20640: 20639: 20628: 20624: 20620: 20617: 20614: 20609: 20605: 20601: 20598: 20594: 20589: 20584: 20580: 20577: 20574: 20569: 20565: 20558: 20554: 20549: 20545: 20541: 20538: 20535: 20530: 20526: 20507: 20506: 20495: 20492: 20488: 20483: 20478: 20474: 20471: 20468: 20463: 20459: 20452: 20448: 20443: 20439: 20435: 20432: 20429: 20424: 20420: 20377: 20355: 20351: 20324: 20320: 20316: 20313: 20289: 20267: 20263: 20259: 20254: 20250: 20227: 20223: 20200: 20196: 20179: 20176: 20142: 20141:Cousin problem 20139: 20120: 20114: 20100: 20099: 20096: 20082: 20077: 20055: 20052: 20049: 20044: 20031: 20020: 20015: 20009: 20003: 20000: 19997: 19975: 19969: 19944: 19939: 19909: 19904: 19896: 19890: 19885: 19867: 19864: 19863: 19862: 19848: 19842: 19817: 19806:of sheaves of 19795: 19792: 19787: 19782: 19775: 19769: 19763: 19758: 19753: 19746: 19740: 19734: 19729: 19724: 19717: 19711: 19705: 19702: 19669: 19663: 19649: 19648: 19636: 19614: 19608: 19583: 19578: 19571: 19566: 19561: 19556: 19549: 19546: 19541: 19535: 19529: 19526: 19506: 19503: 19500: 19480: 19477: 19474: 19463: 19451: 19429: 19424: 19417: 19412: 19407: 19402: 19395: 19392: 19387: 19381: 19358: 19338: 19315: 19293: 19287: 19258: 19231: 19209: 19204: 19198: 19192: 19189: 19186: 19175:coherent sheaf 19159: 19139: 19128: 19127: 19116: 19113: 19108: 19103: 19096: 19091: 19086: 19081: 19074: 19071: 19066: 19060: 19054: 19049: 19044: 19037: 19034: 19029: 19023: 19007:exact sequence 18994: 18974: 18948: 18942: 18917: 18895: 18890: 18884: 18878: 18875: 18872: 18853: 18850: 18848: 18847:Coherent sheaf 18845: 18816: 18813: 18810: 18807: 18804: 18793: 18792: 18781: 18778: 18775: 18772: 18769: 18766: 18762: 18759: 18755: 18752: 18749: 18745: 18742: 18738: 18735: 18732: 18712: 18709: 18706: 18703: 18700: 18696: 18693: 18689: 18685: 18682: 18678: 18675: 18672: 18669: 18666: 18663: 18660: 18649: 18637: 18634: 18631: 18628: 18625: 18621: 18618: 18614: 18611: 18608: 18605: 18602: 18599: 18579: 18575: 18572: 18568: 18565: 18562: 18542: 18539: 18536: 18533: 18530: 18527: 18524: 18521: 18518: 18494: 18474: 18454: 18451: 18448: 18445: 18442: 18422: 18419: 18416: 18404: 18401: 18392: 18389: 18369: 18366: 18352:Lars Hörmander 18337: 18332: 18302: 18299: 18296: 18276: 18273: 18270: 18250: 18247: 18244: 18224: 18221: 18218: 18215: 18212: 18201: 18200: 18194: 18188: 18174: 18168: 18147: 18142: 18137: 18134: 18122: 18119: 18106: 18086: 18064: 18059: 18027: 18021: 17998: 17995: 17992: 17987: 17984: 17980: 17959: 17956: 17953: 17933: 17930: 17927: 17924: 17921: 17893: 17890: 17887: 17855: 17852: 17849: 17846: 17834: 17831: 17816: 17811: 17781: 17778: 17775: 17772: 17769: 17766: 17754: 17751: 17734: 17731: 17728: 17725: 17722: 17719: 17699: 17696: 17693: 17690: 17687: 17681: 17678: 17675: 17672: 17669: 17666: 17646: 17643: 17640: 17636: 17632: 17628: 17624: 17621: 17618: 17615: 17612: 17609: 17604: 17600: 17596: 17591: 17587: 17583: 17580: 17577: 17574: 17571: 17568: 17544: 17541: 17538: 17534: 17530: 17526: 17522: 17519: 17516: 17513: 17510: 17507: 17502: 17498: 17494: 17489: 17485: 17481: 17478: 17475: 17472: 17469: 17466: 17446: 17423: 17420: 17413: 17409: 17405: 17378: 17375: 17371: 17367: 17363: 17342: 17339: 17336: 17333: 17330: 17327: 17324: 17321: 17317: 17313: 17309: 17305: 17302: 17282: 17279: 17276: 17273: 17270: 17265: 17261: 17257: 17252: 17248: 17244: 17241: 17225: 17222: 17220: 17217: 17200: 17180: 17156: 17134: 17105: 17093: 17090: 17085: 17082: 17067: 17061: 17038: 17027: 17026: 17015: 17010: 17006: 17000: 16995: 16992: 16989: 16985: 16981: 16978: 16948: 16942: 16919: 16916: 16911: 16907: 16869: 16863: 16845: 16844: 16833: 16830: 16824: 16819: 16815: 16806: 16802: 16792: 16787: 16783: 16776: 16770: 16766: 16762: 16757: 16754: 16751: 16748: 16743: 16739: 16730: 16725: 16722: 16719: 16716: 16713: 16709: 16705: 16702: 16699: 16696: 16693: 16683: 16671: 16668: 16663: 16659: 16650: 16646: 16642: 16637: 16634: 16631: 16628: 16625: 16617: 16612: 16609: 16606: 16602: 16598: 16595: 16592: 16589: 16586: 16583: 16580: 16557: 16537: 16534: 16531: 16528: 16504: 16501: 16498: 16495: 16492: 16489: 16486: 16483: 16463: 16460: 16457: 16454: 16451: 16448: 16445: 16423: 16417: 16393: 16389: 16384: 16379: 16374: 16371: 16345: 16339: 16324: 16321: 16319: 16316: 16302: 16299: 16279: 16276: 16273: 16268: 16262: 16256: 16253: 16250: 16247: 16239: 16236: 16208: 16205: 16192: 16189: 16186: 16181: 16175: 16169: 16166: 16163: 16160: 16152: 16149: 16121: 16118: 16115: 16110: 16104: 16098: 16095: 16092: 16089: 16081: 16078: 16046: 16043: 16040: 16037: 16034: 16031: 16028: 16025: 16022: 16019: 16016: 16013: 16010: 15986: 15950: 15944: 15938: 15935: 15923: 15920: 15901: 15895: 15876: 15873: 15856: 15850: 15847: 15841: 15836: 15833: 15805: 15799: 15768: 15761: 15758: 15751: 15745: 15741: 15737: 15732: 15727: 15723: 15716: 15711: 15708: 15704: 15700: 15697: 15692: 15689: 15685: 15681: 15678: 15673: 15669: 15645: 15623: 15617: 15594: 15574: 15571: 15567: 15559: 15556: 15551: 15547: 15544: 15539: 15534: 15530: 15523: 15519: 15516: 15513: 15489: 15486: 15483: 15480: 15477: 15474: 15450: 15446: 15443: 15432: 15431: 15420: 15417: 15414: 15411: 15408: 15404: 15400: 15397: 15394: 15391: 15388: 15385: 15362: 15359: 15356: 15353: 15350: 15340: 15339: 15333: 15322: 15319: 15316: 15313: 15310: 15306: 15302: 15299: 15296: 15293: 15283: 15268: 15252: 15251: 15240: 15237: 15234: 15231: 15228: 15225: 15222: 15219: 15216: 15212: 15208: 15205: 15202: 15191: 15190: 15178: 15175: 15172: 15169: 15166: 15163: 15160: 15157: 15154: 15142: 15141: 15128: 15123: 15118: 15115: 15112: 15109: 15087: 15082: 15077: 15074: 15070: 15066: 15063: 15060: 15057: 15054: 15051: 15048: 15045: 15023: 15022: 15009: 15003: 14997: 14994: 14980: 14969: 14966: 14963: 14960: 14957: 14954: 14949: 14944: 14941: 14938: 14935: 14925: 14920: 14917: 14888: 14883: 14879: 14875: 14871: 14867: 14864: 14861: 14841: 14836: 14832: 14828: 14825: 14822: 14818: 14812: 14808: 14803: 14782: 14779: 14774: 14770: 14747: 14742: 14720: 14691: 14670: 14667: 14654: 14649: 14644: 14639: 14636: 14633: 14613: 14610: 14605: 14600: 14595: 14590: 14586: 14565: 14560: 14556: 14552: 14549: 14546: 14513: 14508: 14504: 14500: 14495: 14488: 14485: 14478: 14470: 14467: 14462: 14458: 14454: 14451: 14448: 14415: 14412: 14409: 14386: 14383: 14380: 14377: 14374: 14348: 14341: 14338: 14314: 14294: 14291: 14288: 14255: 14248: 14245: 14238: 14235: 14211: 14184: 14181: 14178: 14173: 14156: 14155: 14144: 14140: 14136: 14133: 14130: 14127: 14122: 14117: 14114: 14105: 14101: 14098: 14095: 14092: 14088: 14082: 14079: 14076: 14072: 14068: 14064: 14060: 14057: 14054: 14051: 14047: 14043: 14040: 14037: 14034: 14030: 14026: 14021: 14014: 14011: 13976: 13973: 13970: 13946: 13943: 13940: 13935: 13924:. Further let 13909: 13889: 13867: 13862: 13857: 13854: 13842: 13839: 13838: 13837: 13825: 13822: 13819: 13808:Cousin problem 13804: 13790: 13786: 13782: 13777: 13773: 13750: 13746: 13723: 13719: 13707: 13689: 13685: 13679: 13674: 13671: 13668: 13664: 13660: 13657: 13637: 13634: 13629: 13625: 13621: 13616: 13612: 13600: 13586: 13582: 13576: 13571: 13568: 13565: 13561: 13557: 13554: 13532: 13528: 13524: 13521: 13518: 13513: 13509: 13495: 13492: 13466: 13462: 13459: 13439: 13436: 13433: 13406: 13403: 13400: 13364: 13361: 13358: 13355: 13352: 13332: 13329: 13326: 13304: 13299: 13294: 13291: 13271: 13268: 13265: 13239: 13234: 13201: 13198: 13195: 13192: 13189: 13181: 13176: 13171: 13168: 13124: 13121: 13095: 13090: 13085: 13082: 13051: 13035: 13032: 12999: 12994: 12970: 12965: 12937: 12932: 12917: 12914: 12913: 12912: 12898: 12894: 12890: 12887: 12882: 12878: 12874: 12871: 12851: 12831: 12828: 12825: 12820: 12816: 12812: 12807: 12804: 12801: 12798: 12794: 12788: 12784: 12780: 12775: 12771: 12748: 12743: 12738: 12733: 12728: 12723: 12720: 12698: 12694: 12671: 12667: 12643: 12640: 12639: 12638: 12626: 12623: 12620: 12617: 12614: 12611: 12608: 12605: 12601: 12598: 12595: 12589: 12586: 12580: 12575: 12571: 12567: 12562: 12557: 12552: 12548: 12541: 12536: 12531: 12526: 12523: 12520: 12517: 12514: 12511: 12508: 12498: 12486: 12483: 12480: 12475: 12470: 12465: 12461: 12457: 12452: 12447: 12442: 12438: 12431: 12426: 12421: 12416: 12413: 12410: 12407: 12404: 12401: 12398: 12388: 12376: 12373: 12370: 12366: 12362: 12358: 12351: 12348: 12345: 12341: 12337: 12333: 12326: 12321: 12316: 12311: 12308: 12305: 12302: 12299: 12296: 12293: 12271: 12268: 12253: 12249: 12226: 12222: 12199: 12195: 12172: 12168: 12145: 12141: 12118: 12114: 12102: 12101: 12093: 12092: 12091: 12041: 12036: 11995: 11994: 11983: 11980: 11977: 11974: 11971: 11968: 11965: 11959: 11955: 11949: 11945: 11940: 11936: 11933: 11930: 11927: 11921: 11915: 11912: 11908: 11902: 11898: 11893: 11889: 11884: 11880: 11876: 11873: 11868: 11864: 11860: 11855: 11851: 11847: 11844: 11841: 11838: 11835: 11830: 11826: 11813: 11812: 11800: 11797: 11794: 11791: 11788: 11768: 11765: 11762: 11758: 11752: 11748: 11743: 11739: 11736: 11733: 11729: 11723: 11719: 11714: 11710: 11705: 11700: 11695: 11692: 11689: 11686: 11681: 11677: 11650: 11645: 11631: 11628: 11626: 11623: 11610: 11607: 11604: 11582: 11578: 11574: 11571: 11568: 11563: 11559: 11531: 11527: 11522: 11516: 11512: 11508: 11503: 11499: 11495: 11492: 11485: 11481: 11476: 11470: 11466: 11462: 11457: 11453: 11449: 11442: 11438: 11434: 11431: 11428: 11423: 11419: 11414: 11408: 11403: 11400: 11395: 11391: 11387: 11384: 11381: 11376: 11372: 11367: 11344: 11339: 11312: 11307: 11290: 11289: 11278: 11274: 11268: 11264: 11259: 11255: 11252: 11249: 11246: 11243: 11239: 11233: 11229: 11224: 11220: 11217: 11214: 11211: 11208: 11205: 11202: 11199: 11196: 11193: 11190: 11187: 11173: 11172: 11161: 11158: 11155: 11150: 11146: 11142: 11139: 11136: 11131: 11127: 11123: 11120: 11117: 11114: 11109: 11105: 11101: 11098: 11095: 11090: 11086: 11082: 11079: 11076: 11073: 11070: 11065: 11061: 11037: 11032: 11028: 11024: 11021: 10998: 10995: 10967: 10966: 10955: 10951: 10947: 10944: 10941: 10938: 10935: 10932: 10929: 10923: 10919: 10913: 10909: 10905: 10900: 10895: 10891: 10886: 10882: 10878: 10872: 10868: 10864: 10859: 10855: 10850: 10846: 10843: 10838: 10834: 10830: 10827: 10824: 10819: 10815: 10811: 10808: 10805: 10801: 10777: 10774: 10769: 10765: 10745: 10744: 10733: 10729: 10725: 10722: 10719: 10716: 10713: 10710: 10704: 10699: 10696: 10692: 10687: 10683: 10680: 10675: 10671: 10666: 10662: 10659: 10656: 10653: 10650: 10647: 10642: 10638: 10634: 10631: 10628: 10623: 10619: 10615: 10612: 10609: 10605: 10581: 10578: 10573: 10569: 10544: 10540: 10535: 10531: 10527: 10522: 10517: 10513: 10509: 10502: 10498: 10494: 10490: 10485: 10481: 10477: 10471: 10467: 10463: 10458: 10454: 10449: 10424: 10421: 10418: 10415: 10412: 10409: 10406: 10403: 10400: 10394: 10390: 10371: 10370: 10359: 10355: 10351: 10348: 10345: 10342: 10339: 10336: 10333: 10327: 10323: 10317: 10313: 10309: 10304: 10299: 10295: 10290: 10286: 10282: 10276: 10272: 10268: 10263: 10259: 10254: 10250: 10247: 10242: 10238: 10234: 10231: 10228: 10223: 10219: 10215: 10212: 10209: 10205: 10181: 10178: 10175: 10170: 10165: 10161: 10157: 10154: 10151: 10146: 10141: 10137: 10133: 10130: 10125: 10121: 10098: 10093: 10088: 10085: 10080: 10076: 10072: 10069: 10066: 10061: 10057: 10053: 10050: 10047: 10027: 10024: 10021: 9999: 9994: 9989: 9986: 9970: 9967: 9942: 9939: 9936: 9915: 9912: 9892: 9889: 9886: 9881: 9876: 9873: 9853: 9850: 9847: 9842: 9837: 9834: 9810: 9782: 9779: 9733: 9728: 9723: 9720: 9715: 9710: 9705: 9685: 9682: 9679: 9655: 9635: 9632: 9629: 9626: 9623: 9617: 9614: 9608: 9605: 9581: 9578: 9575: 9570: 9548: 9545: 9542: 9537: 9532: 9529: 9509: 9506: 9503: 9498: 9493: 9490: 9468: 9463: 9445: 9442: 9409: 9406: 9403: 9391: 9388: 9359: 9339: 9336: 9333: 9330: 9325: 9322: 9318: 9297: 9294: 9291: 9288: 9285: 9263: 9258: 9229: 9228:Biholomorphism 9226: 9198: 9195: 9190: 9186: 9163: 9158: 9134: 9130: 9126: 9123: 9118: 9114: 9110: 9105: 9101: 9097: 9094: 9074: 9071: 9068: 9063: 9059: 9055: 9050: 9046: 9042: 9039: 9005: 9000: 8995: 8992: 8976: 8973: 8972: 8971: 8959: 8956: 8953: 8950: 8947: 8944: 8941: 8938: 8935: 8932: 8929: 8924: 8921: 8917: 8913: 8910: 8907: 8904: 8901: 8882: 8881: 8869: 8866: 8863: 8860: 8857: 8854: 8851: 8848: 8845: 8842: 8839: 8836: 8831: 8828: 8821: 8817: 8813: 8810: 8807: 8804: 8801: 8798: 8795: 8792: 8789: 8786: 8783: 8778: 8775: 8771: 8767: 8764: 8761: 8758: 8755: 8744: 8731: 8727: 8723: 8720: 8715: 8709: 8706: 8700: 8697: 8694: 8691: 8686: 8682: 8678: 8675: 8672: 8669: 8664: 8660: 8656: 8653: 8648: 8642: 8639: 8633: 8629: 8624: 8618: 8615: 8609: 8604: 8598: 8595: 8589: 8584: 8581: 8578: 8575: 8572: 8568: 8559: 8556: 8551: 8546: 8543: 8540: 8536: 8531: 8521: 8517: 8513: 8510: 8507: 8504: 8499: 8496: 8493: 8490: 8487: 8484: 8478: 8475: 8472: 8469: 8466: 8463: 8460: 8437: 8434: 8431: 8428: 8425: 8422: 8419: 8399: 8375: 8372: 8369: 8366: 8363: 8360: 8338: 8333: 8307: 8275: 8255: 8252: 8230: 8225: 8196: 8193: 8178: 8174: 8170: 8166: 8162: 8158: 8135: 8131: 8127: 8123: 8119: 8115: 8093: 8089: 8085: 8081: 8077: 8073: 8069: 8065: 8061: 8057: 8053: 8041: 8040: 8025: 8022: 8019: 8014: 8010: 8006: 8003: 8000: 7995: 7991: 7987: 7977: 7973: 7969: 7964: 7959: 7955: 7951: 7947: 7943: 7940: 7937: 7932: 7929: 7924: 7920: 7914: 7910: 7906: 7901: 7898: 7893: 7889: 7883: 7879: 7875: 7868: 7863: 7859: 7855: 7852: 7847: 7843: 7837: 7834: 7827: 7824: 7821: 7818: 7814: 7810: 7807: 7804: 7801: 7798: 7795: 7792: 7785: 7781: 7777: 7773: 7767: 7763: 7758: 7753: 7746: 7743: 7740: 7736: 7728: 7725: 7721: 7714: 7709: 7706: 7703: 7699: 7695: 7692: 7690: 7688: 7683: 7679: 7675: 7670: 7666: 7662: 7657: 7654: 7651: 7646: 7639: 7636: 7633: 7629: 7619: 7615: 7611: 7605: 7601: 7594: 7589: 7586: 7583: 7580: 7577: 7574: 7571: 7564: 7560: 7556: 7552: 7546: 7542: 7537: 7532: 7523: 7519: 7515: 7512: 7509: 7506: 7502: 7494: 7491: 7487: 7480: 7475: 7472: 7469: 7465: 7461: 7458: 7456: 7454: 7451: 7448: 7445: 7442: 7441: 7417: 7413: 7410: 7407: 7404: 7401: 7398: 7395: 7387: 7382: 7378: 7374: 7370: 7366: 7362: 7358: 7353: 7349: 7345: 7340: 7335: 7330: 7327: 7322: 7318: 7314: 7311: 7308: 7303: 7299: 7295: 7290: 7286: 7282: 7279: 7276: 7272: 7248: 7245: 7242: 7239: 7227: 7224: 7206: 7202: 7199: 7196: 7193: 7190: 7187: 7184: 7176: 7171: 7167: 7163: 7159: 7153: 7149: 7145: 7140: 7136: 7131: 7127: 7122: 7117: 7112: 7109: 7104: 7100: 7096: 7093: 7090: 7085: 7081: 7077: 7072: 7068: 7064: 7061: 7058: 7054: 7032: 7028: 7025: 7022: 7019: 7016: 7013: 7010: 7002: 6997: 6993: 6989: 6985: 6979: 6975: 6971: 6966: 6962: 6957: 6953: 6948: 6943: 6938: 6935: 6930: 6926: 6922: 6919: 6916: 6911: 6907: 6903: 6898: 6894: 6890: 6887: 6884: 6880: 6852: 6848: 6843: 6837: 6833: 6829: 6824: 6820: 6816: 6813: 6806: 6802: 6797: 6791: 6787: 6783: 6778: 6774: 6770: 6763: 6759: 6755: 6752: 6749: 6744: 6740: 6735: 6729: 6724: 6721: 6716: 6712: 6708: 6705: 6702: 6697: 6693: 6688: 6667: 6664: 6661: 6658: 6655: 6652: 6649: 6646: 6643: 6640: 6632: 6628: 6624: 6612: 6609: 6608: 6607: 6589: 6585: 6577: 6573: 6567: 6564: 6557: 6553: 6545: 6541: 6535: 6528: 6524: 6516: 6512: 6508: 6505: 6502: 6497: 6493: 6488: 6479: 6474: 6471: 6468: 6464: 6460: 6450: 6446: 6438: 6434: 6428: 6425: 6418: 6414: 6406: 6402: 6396: 6391: 6384: 6380: 6376: 6373: 6370: 6365: 6361: 6356: 6341: 6340: 6318: 6314: 6287: 6283: 6252: 6248: 6244: 6241: 6238: 6233: 6229: 6212: 6211: 6200: 6190: 6186: 6181: 6175: 6171: 6167: 6162: 6158: 6154: 6151: 6144: 6140: 6135: 6129: 6125: 6121: 6116: 6112: 6108: 6101: 6097: 6093: 6090: 6087: 6082: 6078: 6073: 6067: 6062: 6059: 6054: 6050: 6046: 6043: 6040: 6035: 6031: 6026: 6022: 6019: 6016: 6013: 6010: 5983: 5980: 5977: 5974: 5952: 5947: 5942: 5939: 5936: 5933: 5928: 5924: 5920: 5917: 5914: 5909: 5905: 5901: 5898: 5895: 5877: 5876: 5859: 5855: 5851: 5848: 5843: 5839: 5835: 5826: 5823: 5818: 5814: 5809: 5803: 5799: 5795: 5790: 5786: 5782: 5779: 5774: 5771: 5766: 5762: 5757: 5751: 5747: 5743: 5738: 5734: 5730: 5725: 5720: 5716: 5712: 5709: 5706: 5701: 5697: 5693: 5690: 5680: 5676: 5672: 5668: 5664: 5657: 5653: 5649: 5645: 5636: 5632: 5628: 5625: 5622: 5619: 5615: 5610: 5603: 5599: 5595: 5590: 5586: 5581: 5577: 5575: 5572: 5562: 5558: 5553: 5547: 5543: 5539: 5534: 5530: 5526: 5523: 5516: 5512: 5507: 5501: 5497: 5493: 5488: 5484: 5480: 5473: 5469: 5465: 5462: 5459: 5454: 5450: 5445: 5439: 5434: 5431: 5426: 5422: 5418: 5415: 5412: 5407: 5403: 5398: 5394: 5391: 5388: 5385: 5382: 5379: 5377: 5354: 5351: 5348: 5345: 5342: 5339: 5336: 5333: 5325: 5320: 5316: 5312: 5308: 5302: 5298: 5294: 5289: 5285: 5280: 5276: 5271: 5266: 5261: 5258: 5253: 5249: 5245: 5242: 5239: 5234: 5230: 5226: 5221: 5217: 5213: 5210: 5207: 5204: 5188: 5185: 5177: 5176: 5158: 5154: 5146: 5142: 5136: 5129: 5125: 5117: 5113: 5105: 5100: 5096: 5092: 5087: 5083: 5079: 5073: 5069: 5059: 5055: 5047: 5043: 5037: 5034: 5027: 5023: 5015: 5011: 5007: 5000: 4995: 4991: 4987: 4984: 4981: 4976: 4972: 4968: 4963: 4959: 4955: 4952: 4945: 4941: 4937: 4934: 4931: 4926: 4922: 4917: 4910: 4885: 4881: 4877: 4873: 4869: 4847: 4843: 4840: 4837: 4834: 4831: 4828: 4825: 4817: 4812: 4808: 4804: 4800: 4794: 4790: 4786: 4781: 4777: 4772: 4768: 4763: 4758: 4753: 4750: 4745: 4741: 4737: 4734: 4731: 4726: 4722: 4718: 4713: 4709: 4705: 4702: 4699: 4695: 4665: 4659: 4640: 4639: 4630: 4628: 4617: 4612: 4608: 4604: 4601: 4596: 4592: 4588: 4579: 4576: 4571: 4567: 4562: 4556: 4552: 4548: 4543: 4539: 4535: 4532: 4527: 4524: 4519: 4515: 4510: 4504: 4500: 4496: 4491: 4487: 4483: 4478: 4473: 4469: 4465: 4462: 4459: 4454: 4450: 4446: 4443: 4433: 4429: 4425: 4421: 4417: 4410: 4406: 4402: 4398: 4389: 4385: 4381: 4378: 4375: 4372: 4367: 4362: 4358: 4354: 4349: 4345: 4338: 4328: 4324: 4316: 4312: 4306: 4303: 4296: 4292: 4284: 4280: 4274: 4269: 4264: 4260: 4256: 4253: 4250: 4245: 4241: 4237: 4232: 4228: 4224: 4221: 4214: 4210: 4206: 4203: 4200: 4195: 4191: 4186: 4159: 4156: 4153: 4152: 4143: 4141: 4128: 4124: 4120: 4117: 4112: 4108: 4104: 4097: 4092: 4088: 4084: 4079: 4075: 4071: 4068: 4065: 4060: 4056: 4052: 4047: 4043: 4039: 4034: 4029: 4025: 4021: 4018: 4015: 4010: 4006: 4002: 3999: 3989: 3985: 3981: 3977: 3973: 3966: 3962: 3958: 3954: 3945: 3941: 3937: 3934: 3931: 3928: 3924: 3919: 3916: 3911: 3907: 3903: 3900: 3897: 3892: 3888: 3884: 3881: 3844: 3841: 3830: 3829: 3812: 3808: 3804: 3797: 3792: 3788: 3784: 3779: 3775: 3771: 3768: 3765: 3760: 3756: 3752: 3747: 3743: 3739: 3736: 3731: 3727: 3723: 3718: 3714: 3710: 3705: 3700: 3696: 3692: 3689: 3686: 3681: 3677: 3673: 3668: 3664: 3660: 3657: 3647: 3643: 3639: 3635: 3629: 3625: 3621: 3613: 3609: 3605: 3601: 3597: 3592: 3588: 3584: 3576: 3572: 3568: 3564: 3555: 3551: 3547: 3544: 3541: 3538: 3534: 3529: 3526: 3524: 3522: 3517: 3513: 3509: 3502: 3497: 3493: 3489: 3484: 3480: 3476: 3473: 3468: 3464: 3460: 3455: 3451: 3447: 3442: 3437: 3433: 3429: 3426: 3423: 3418: 3414: 3410: 3405: 3401: 3397: 3392: 3388: 3384: 3381: 3371: 3367: 3363: 3359: 3353: 3349: 3345: 3337: 3333: 3329: 3325: 3316: 3312: 3308: 3305: 3302: 3299: 3295: 3290: 3287: 3285: 3283: 3278: 3274: 3270: 3261: 3257: 3253: 3248: 3244: 3238: 3233: 3229: 3225: 3222: 3219: 3214: 3210: 3206: 3201: 3197: 3193: 3190: 3180: 3176: 3172: 3168: 3161: 3158: 3155: 3151: 3146: 3143: 3141: 3139: 3134: 3130: 3126: 3123: 3120: 3115: 3111: 3107: 3104: 3101: 3100: 3071: 3066: 3063: 3060: 3056: 3052: 3049: 3028: 3024: 3021: 3018: 3015: 3012: 3009: 3006: 2996: 2992: 2988: 2984: 2978: 2974: 2970: 2965: 2961: 2956: 2952: 2947: 2942: 2937: 2934: 2929: 2925: 2921: 2918: 2915: 2910: 2906: 2902: 2897: 2893: 2889: 2886: 2883: 2879: 2875: 2872: 2869: 2866: 2863: 2860: 2855: 2852: 2827: 2823: 2819: 2816: 2813: 2808: 2804: 2800: 2795: 2791: 2786: 2781: 2778: 2754: 2751: 2726: 2723: 2718: 2712: 2708: 2704: 2701: 2698: 2693: 2689: 2685: 2680: 2676: 2650: 2644: 2640: 2636: 2633: 2630: 2625: 2621: 2617: 2612: 2608: 2582: 2578: 2555: 2551: 2530: 2527: 2524: 2521: 2518: 2515: 2512: 2509: 2506: 2484: 2478: 2449: 2445: 2424: 2401: 2398: 2386: 2383: 2380: 2374: 2371: 2344: 2341: 2338: 2330: 2325: 2321: 2315: 2310: 2307: 2300: 2297: 2294: 2291: 2288: 2285: 2282: 2279: 2276: 2273: 2270: 2239: 2235: 2231: 2226: 2223: 2217: 2214: 2206: 2202: 2198: 2193: 2190: 2172: 2168: 2164: 2159: 2156: 2150: 2142: 2138: 2134: 2129: 2126: 2119: 2116: 2113: 2110: 2107: 2104: 2101: 2098: 2095: 2092: 2089: 2069: 2049: 2029: 2008: 2004: 1999: 1994: 1989: 1986: 1964: 1961: 1958: 1952: 1949: 1944: 1941: 1935: 1932: 1929: 1926: 1923: 1917: 1914: 1909: 1906: 1893: 1890: 1887: 1881: 1878: 1873: 1870: 1864: 1861: 1858: 1855: 1849: 1846: 1841: 1838: 1815: 1791: 1771: 1750: 1746: 1743: 1722: 1718: 1714: 1710: 1707: 1695: 1692: 1680:Osgood's lemma 1649: 1644: 1640: 1636: 1633: 1630: 1625: 1622: 1619: 1615: 1611: 1608: 1605: 1600: 1597: 1594: 1590: 1586: 1583: 1580: 1575: 1571: 1567: 1564: 1561: 1558: 1519: 1516: 1513: 1510: 1507: 1504: 1501: 1498: 1495: 1492: 1489: 1486: 1483: 1480: 1477: 1474: 1471: 1468: 1465: 1462: 1459: 1456: 1453: 1430: 1426: 1421: 1416: 1411: 1408: 1388: 1385: 1382: 1361: 1338: 1333: 1328: 1325: 1309: 1306: 1304: 1301: 1286: 1281: 1257: 1252: 1219: 1218: 1207: 1202: 1197: 1192: 1188: 1184: 1179: 1175: 1171: 1166: 1162: 1144: 1143: 1132: 1127: 1121: 1118: 1116: 1113: 1112: 1109: 1106: 1103: 1101: 1098: 1097: 1095: 1062:multiplication 1040:imaginary unit 1036:multiplication 983: 980: 975: 948: 943: 916: 867: 862: 849:Stein manifold 834: 829: 805: 800: 777: 744: 739: 722: 719: 715:Banach algebra 550: 545: 530:. In fact the 511: 436: 432: 427: 422: 417: 414: 304: 301: 278: 273: 249: 244: 241: 211: 208: 205: 199: 194: 189: 184: 181: 156: 152: 149: 118:uniform limits 112: 75:analytic space 48: 43: 21:The theory of 15: 9: 6: 4: 3: 2: 31368: 31357: 31354: 31352: 31349: 31348: 31346: 31337: 31335: 31331: 31327: 31323: 31319: 31315: 31311: 31307: 31303: 31299: 31294: 31291: 31287: 31283: 31279: 31275: 31269: 31265: 31258: 31253: 31250: 31246: 31243: 31240: 31237: 31236: 31225: 31221: 31217: 31210: 31205: 31200: 31195: 31191: 31187: 31183: 31178: 31173: 31168: 31164: 31160: 31156: 31151: 31146: 31141: 31138:(2): 79–102. 31137: 31133: 31129: 31124: 31121: 31115: 31111: 31106: 31102: 31098: 31094: 31090: 31086: 31082: 31077: 31074: 31070: 31066: 31062: 31058: 31054: 31049: 31045: 31041: 31037: 31033: 31029: 31025: 31021: 31015: 31011: 31007: 31003: 30999: 30995: 30991: 30990: 30979: 30975: 30974: 30969: 30964: 30961: 30957: 30956: 30951: 30946: 30943: 30939: 30938: 30933: 30928: 30925: 30921: 30920: 30915: 30910: 30907: 30903: 30902: 30897: 30892: 30889: 30885: 30884: 30879: 30874: 30871: 30867: 30866: 30861: 30856: 30853: 30849: 30848: 30843: 30838: 30835: 30831: 30830: 30825: 30820: 30817: 30813: 30812: 30807: 30802: 30799: 30795: 30794: 30789: 30784: 30781: 30777: 30776: 30771: 30766: 30763: 30759: 30758: 30753: 30748: 30745: 30741: 30740: 30735: 30730: 30727: 30723: 30722: 30717: 30712: 30709: 30705: 30704: 30699: 30694: 30693: 30681: 30679:9780486458120 30675: 30671: 30670: 30664: 30660: 30656: 30652: 30646: 30642: 30638: 30634: 30629: 30625: 30623:3-7643-7490-X 30619: 30615: 30611: 30607: 30603: 30599: 30593: 30589: 30585: 30581: 30577: 30573: 30567: 30563: 30559: 30555: 30550: 30546: 30544:3-540-22614-1 30540: 30536: 30532: 30528: 30527: 30521: 30517: 30515:9783110838350 30511: 30507: 30506: 30500: 30496: 30490: 30486: 30485: 30479: 30476: 30470: 30466: 30465: 30460: 30456: 30450: 30448:9780821819753 30444: 30440: 30436: 30432: 30428: 30427: 30424: 30420: 30416: 30411: 30408: 30404: 30400: 30398:3-540-90388-7 30394: 30390: 30385: 30381: 30375: 30371: 30367: 30363: 30358: 30354: 30348: 30344: 30340: 30336: 30335: 30330: 30326: 30320: 30318:9780486318677 30314: 30310: 30309: 30303: 30302: 30299: 30297:9782705652159 30293: 30289: 30284: 30278: 30272: 30268: 30263: 30262: 30259: 30255: 30251: 30246: 30242: 30240:0-387-90617-7 30236: 30232: 30228: 30227:Forster, Otto 30224: 30220: 30214: 30210: 30205: 30201: 30195: 30191: 30187: 30183: 30178: 30177: 30163: 30159: 30154: 30149: 30145: 30141: 30137: 30130: 30121: 30116: 30112: 30108: 30104: 30097: 30089: 30085: 30080: 30075: 30071: 30067: 30063: 30056: 30048: 30044: 30040: 30036: 30032: 30028: 30021: 30013: 30009: 30005: 30001: 29997: 29993: 29986: 29978: 29974: 29970: 29966: 29962: 29958: 29951: 29943: 29939: 29934: 29929: 29925: 29921: 29917: 29910: 29902: 29896: 29892: 29888: 29884: 29877: 29869: 29863: 29859: 29855: 29851: 29844: 29836: 29832: 29828: 29824: 29820: 29816: 29812: 29805: 29797: 29791: 29787: 29783: 29779: 29772: 29764: 29760: 29756: 29752: 29748: 29741: 29734: 29730: 29726: 29722: 29718: 29714: 29710: 29706: 29702: 29698: 29692: 29684: 29680: 29675: 29670: 29666: 29662: 29657: 29652: 29648: 29644: 29640: 29636: 29632: 29625: 29617: 29613: 29609: 29605: 29601: 29597: 29590: 29582: 29578: 29574: 29570: 29566: 29562: 29555: 29547: 29543: 29539: 29535: 29531: 29527: 29523: 29517: 29513: 29509: 29505: 29501: 29500: 29495: 29489: 29481: 29475: 29471: 29467: 29463: 29462: 29454: 29446: 29444:9781461418092 29440: 29436: 29435: 29427: 29419: 29417:9780821802687 29413: 29409: 29405: 29401: 29400: 29392: 29384: 29380: 29376: 29372: 29368: 29364: 29357: 29348: 29343: 29339: 29335: 29331: 29324: 29317: 29313: 29312: 29307: 29300: 29292: 29288: 29283: 29278: 29274: 29270: 29269: 29264: 29257: 29248: 29243: 29239: 29235: 29231: 29224: 29216: 29212: 29208: 29205:(in French). 29204: 29200: 29193: 29185: 29181: 29177: 29173: 29169: 29165: 29158: 29150: 29146: 29142: 29138: 29134: 29130: 29123: 29115: 29111: 29106: 29101: 29097: 29093: 29089: 29082: 29074: 29072:9784431568513 29068: 29064: 29060: 29056: 29055: 29047: 29039: 29035: 29031: 29027: 29023: 29019: 29015: 29008: 28999: 28994: 28987: 28985: 28983: 28981: 28972: 28968: 28964: 28960: 28956: 28952: 28945: 28937: 28933: 28929: 28925: 28921: 28917: 28910: 28903: 28899: 28895: 28891: 28887: 28883: 28879: 28875: 28868: 28866: 28864: 28862: 28853: 28849: 28845: 28841: 28837: 28833: 28826: 28818: 28814: 28810: 28806: 28801: 28796: 28792: 28788: 28781: 28779: 28770: 28766: 28762: 28758: 28751: 28742: 28737: 28733: 28729: 28725: 28718: 28709: 28704: 28700: 28696: 28692: 28685: 28677: 28673: 28669: 28665: 28661: 28657: 28653: 28646: 28638: 28634: 28629: 28624: 28620: 28616: 28609: 28602: 28595: 28591: 28587: 28583: 28579: 28575: 28571: 28568:(in German), 28567: 28560: 28552: 28548: 28544: 28540: 28535: 28530: 28526: 28522: 28515: 28507: 28503: 28499: 28493: 28489: 28485: 28480: 28475: 28471: 28467: 28460: 28452: 28448: 28444: 28440: 28436: 28432: 28425: 28418: 28414: 28410: 28406: 28402: 28398: 28394: 28390: 28383: 28381: 28379: 28371: 28367: 28363: 28357: 28353: 28349: 28348: 28343: 28342:Weyl, Hermann 28337: 28335: 28326: 28322: 28318: 28311: 28302: 28297: 28293: 28289: 28285: 28278: 28269: 28264: 28260: 28256: 28252: 28245: 28237: 28231: 28227: 28223: 28216: 28207: 28202: 28198: 28194: 28190: 28183: 28181: 28179: 28169: 28164: 28160: 28156: 28152: 28150: 28141: 28133: 28127: 28123: 28119: 28115: 28108: 28097: 28090: 28082: 28076: 28072: 28071: 28063: 28055: 28049: 28045: 28044: 28036: 28029: 28025: 28021: 28017: 28012: 28007: 28003: 27999: 27992: 27985: 27977: 27975:9784431568513 27971: 27967: 27963: 27959: 27958: 27950: 27942: 27936: 27932: 27928: 27924: 27923: 27915: 27907: 27903: 27899: 27895: 27891: 27887: 27883: 27879: 27875: 27868: 27861: 27857: 27853: 27849: 27845: 27841: 27837: 27833: 27826: 27822: 27816: 27814: 27812: 27803: 27799: 27795: 27789: 27785: 27781: 27777: 27776: 27768: 27760: 27756: 27752: 27746: 27742: 27738: 27734: 27733: 27725: 27717: 27713: 27709: 27705: 27700: 27695: 27691: 27687: 27680: 27678: 27669: 27665: 27661: 27657: 27653: 27637: 27617: 27609: 27605: 27601: 27597: 27590: 27583: 27579: 27575: 27571: 27567: 27563: 27559: 27555: 27548: 27541: 27537: 27533: 27529: 27525: 27521: 27514: 27507: 27503: 27498: 27493: 27489: 27485: 27481: 27477: 27470: 27463: 27459: 27458: 27453: 27446: 27439: 27435: 27431: 27427: 27423: 27419: 27415: 27411: 27407: 27400: 27391: 27386: 27382: 27378: 27374: 27367: 27359: 27355: 27350: 27345: 27341: 27337: 27336: 27331: 27324: 27316: 27312: 27308: 27304: 27300: 27296: 27292: 27285: 27277: 27273: 27269: 27265: 27258: 27256: 27246: 27241: 27237: 27233: 27229: 27200: 27193: 27189: 27188: 27183: 27176: 27167: 27162: 27158: 27154: 27150: 27143: 27141: 27133: 27129: 27124: 27119: 27115: 27111: 27104: 27102: 27100: 27098: 27088: 27083: 27079: 27075: 27071: 27064: 27055: 27050: 27046: 27042: 27038: 27031: 27023: 27021:9780821827246 27017: 27013: 27012: 27004: 26996: 26994:9781468492736 26990: 26986: 26985: 26977: 26971: 26966: 26964: 26962: 26960: 26950: 26945: 26941: 26937: 26933: 26926: 26917: 26912: 26908: 26904: 26900: 26893: 26885: 26881: 26877: 26873: 26869: 26865: 26858: 26850: 26844: 26840: 26836: 26832: 26828: 26821: 26819: 26810: 26806: 26799: 26797: 26788: 26784: 26780: 26776: 26769: 26761: 26755: 26751: 26747: 26743: 26742: 26735: 26727: 26723: 26719: 26715: 26711: 26707: 26703: 26699: 26692: 26690: 26688: 26686: 26684: 26675: 26671: 26664: 26656: 26652: 26648: 26644: 26640: 26633: 26631: 26621: 26616: 26612: 26608: 26604: 26597: 26595: 26593: 26591: 26582: 26577: 26573: 26569: 26561: 26556: 26552: 26548: 26541: 26539: 26530: 26523: 26514: 26509: 26505: 26501: 26497: 26490: 26488: 26486: 26477: 26470: 26459: 26452: 26444: 26440: 26436: 26432: 26427: 26422: 26418: 26414: 26407: 26405: 26403: 26401: 26393: 26389: 26384: 26379: 26375: 26371: 26364: 26357: 26353: 26348: 26343: 26339: 26335: 26328: 26326: 26324: 26316: 26312: 26308: 26304: 26300: 26296: 26295: 26290: 26283: 26281: 26271: 26266: 26262: 26258: 26254: 26247: 26239: 26235: 26230: 26225: 26221: 26217: 26213: 26206: 26198: 26194: 26189: 26184: 26180: 26176: 26172: 26165: 26157: 26153: 26149: 26145: 26144: 26136: 26127: 26122: 26118: 26114: 26110: 26103: 26095: 26091: 26086: 26081: 26076: 26071: 26067: 26063: 26059: 26052: 26044: 26040: 26035: 26030: 26026: 26022: 26021: 26016: 26009: 26000: 25995: 25988: 25986: 25978: 25974: 25970: 25967:(in German), 25966: 25962: 25958: 25952: 25950: 25941: 25937: 25933: 25929: 25925: 25921: 25914: 25906: 25900: 25896: 25892: 25888: 25887: 25879: 25871: 25867: 25862: 25857: 25853: 25849: 25845: 25838: 25830: 25824: 25820: 25816: 25812: 25811: 25803: 25795: 25789: 25785: 25781: 25777: 25776: 25771: 25764: 25756: 25752: 25747: 25742: 25738: 25734: 25730: 25723: 25715: 25713:9780521283014 25709: 25705: 25701: 25697: 25690: 25688: 25679: 25675: 25671: 25667: 25660: 25653: 25649: 25648: 25643: 25636: 25627: 25622: 25618: 25614: 25610: 25603: 25595: 25591: 25587: 25583: 25579: 25575: 25570: 25565: 25561: 25558:(in French). 25557: 25556: 25551: 25547: 25541: 25539: 25537: 25528: 25522: 25518: 25514: 25510: 25503: 25495: 25491: 25487: 25483: 25479: 25475: 25468: 25466: 25464: 25455: 25451: 25447: 25443: 25439: 25435: 25430: 25425: 25421: 25417: 25413: 25406: 25404: 25402: 25400: 25398: 25389: 25385: 25380: 25375: 25371: 25367: 25363: 25356: 25354: 25352: 25350: 25348: 25346: 25344: 25342: 25340: 25331: 25327: 25322: 25317: 25313: 25309: 25305: 25298: 25296: 25287: 25281: 25277: 25276: 25268: 25260: 25256: 25251: 25246: 25242: 25238: 25234: 25198: 25196: 25191: 25136: 25119: 25109: 25102: 25098: 25092: 25071: 25042: 25032: 25027: 25011: 25002: 24993: 24984: 24975: 24966: 24957: 24951: 24945: 24936: 24926: 24917: 24908: 24901: 24883: 24866: 24859: 24856:itself is by 24855: 24849: 24840: 24831: 24824: 24818: 24808: 24787: 24783: 24774: 24768: 24761: 24755: 24746: 24739: 24735: 24729: 24722: 24719: 24716: 24710: 24706: 24696: 24693: 24691: 24688: 24686: 24683: 24681: 24680:Harmonic maps 24678: 24676: 24673: 24671: 24668: 24666: 24663: 24661: 24658: 24657: 24651: 24650:by Takeuchi. 24635: 24616: 24612: 24607: 24603: 24598: 24594: 24590: 24586: 24582: 24579: 24575: 24571: 24567: 24563: 24545: 24527: 24523: 24519: 24515: 24511: 24507: 24506:Serre duality 24503: 24499: 24495: 24477: 24464: 24458: 24420: 24416: 24412: 24402: 24400: 24394: 24392: 24388: 24383: 24381: 24377: 24373: 24365: 24358: 24339: 24336: 24330: 24322: 24319: 24315: 24294: 24288: 24285: 24279: 24271: 24268: 24264: 24255: 24248: 24229: 24221: 24218: 24214: 24210: 24205: 24201: 24192: 24188: 24184: 24180: 24176: 24158: 24155: 24149: 24143: 24140: 24134: 24131: 24128: 24117: 24113: 24097: 24089: 24085: 24081: 24077: 24058: 24055: 24049: 24043: 24040: 24037: 24034: 24031: 24008: 24005: 24002: 23984: 23976: 23972: 23956: 23948: 23944: 23940: 23937: 23934: 23930: 23926: 23910: 23907: 23904: 23896: 23892: 23889: 23888:Runge theorem 23885: 23881: 23877: 23874: 23870: 23866: 23865: 23864: 23862: 23861:ambient space 23854: 23851: 23850:biholomorphic 23833: 23830: 23827: 23824: 23809: 23805: 23801: 23798: 23794: 23777: 23762: 23745: 23730: 23729: 23723: 23703: 23700: 23697: 23694: 23691: 23688: 23685: 23671: 23667: 23660: 23655: 23652: 23648: 23644: 23641: 23638: 23635: 23629: 23626: 23604: 23600: 23574: 23570: 23563: 23560: 23557: 23535: 23520: 23502: 23498: 23494: 23491: 23488: 23483: 23479: 23458: 23455: 23450: 23446: 23437: 23433: 23428: 23420: 23414: 23396: 23392: 23381: 23378: 23374: 23370: 23367: 23359: 23356:Suppose that 23355: 23354: 23353: 23347: 23343: 23327: 23321: 23318: 23315: 23307: 23303: 23300:Suppose that 23299: 23298: 23297: 23292:itself Stein? 23291: 23275: 23272: 23269: 23246: 23240: 23220: 23214: 23211: 23191: 23188: 23185: 23177: 23174:Suppose that 23173: 23172: 23171: 23165: 23164: 23163: 23161: 23153: 23149: 23145: 23141: 23137: 23121: 23118: 23115: 23107: 23106: 23105: 23103: 23099: 23094: 23078: 23059: 23055: 23051: 23047: 23043: 23027: 23024: 23021: 23014: 23010: 23009: 23008: 23003:Levi problems 23000: 22998: 22993: 22979: 22976: 22965: 22962: 22954: 22950: 22929: 22926: 22918: 22906: 22903: 22895: 22891: 22887: 22879: 22867: 22864: 22856: 22852: 22843: 22819: 22807: 22804: 22796: 22792: 22777: 22774: 22766: 22762: 22750: 22745: 22733: 22730: 22722: 22718: 22706: 22694: 22691: 22683: 22679: 22671: 22670: 22669: 22667: 22651: 22648: 22640: 22635: 22623: 22620: 22612: 22608: 22599: 22595: 22591: 22587: 22582: 22580: 22576: 22572: 22568: 22558: 22537: 22519: 22515: 22501: 22495: 22489: 22486: 22480: 22474: 22451: 22438: 22435: 22427: 22411: 22408: 22405: 22397: 22393: 22390: 22387: 22383: 22365: 22361: 22354: 22341: 22338: 22329: 22318: 22312: 22302: 22299: 22296: 22288: 22277: 22271: 22263: 22260: 22257: 22254: 22250: 22246: 22237: 22227: 22226: 22224: 22208: 22205: 22202: 22194: 22191: 22189: 22187: 22183: 22179: 22160: 22130: 22122: 22119: 22115: 22105: 22103: 22098: 22080: 22065: 22061: 22057: 22053: 22049: 22045: 22042:is a complex 22041: 22036: 22020: 22005: 22001: 21983: 21954: 21939: 21935: 21917: 21888: 21885: 21870: 21866: 21862: 21858: 21832: 21828: 21813: 21799: 21796: 21785: 21782: 21774: 21770: 21749: 21746: 21743: 21723: 21720: 21709: 21706: 21698: 21694: 21685: 21666: 21655: 21652: 21644: 21640: 21625: 21622: 21614: 21610: 21606: 21603: 21600: 21589: 21579: 21576: 21568: 21564: 21549: 21546: 21538: 21534: 21526: 21525: 21524: 21502: 21499: 21491: 21487: 21478: 21457: 21454: 21451: 21428: 21420: 21408: 21404: 21387: 21384: 21381: 21375: 21368: 21367: 21366: 21344: 21341: 21333: 21329: 21306: 21274: 21264: 21261: 21253: 21249: 21239: 21226: 21223: 21215: 21205: 21202: 21194: 21190: 21161: 21151: 21148: 21140: 21136: 21124: 21113: 21107: 21097: 21094: 21086: 21082: 21075: 21071: 21061: 21051: 21048: 21040: 21036: 21028: 21027: 21026: 21009: 21001: 20990: 20984: 20974: 20971: 20963: 20959: 20952: 20948: 20938: 20928: 20925: 20917: 20913: 20905: 20904: 20903: 20889: 20867: 20856: 20850: 20835: 20817: 20788: 20772: 20756: 20752: 20747: 20743: 20735: 20719: 20697: 20693: 20688: 20682: 20678: 20663: 20661: 20657: 20653: 20649: 20645: 20618: 20615: 20607: 20603: 20587: 20578: 20575: 20567: 20563: 20556: 20552: 20539: 20536: 20528: 20524: 20516: 20515: 20514: 20512: 20493: 20481: 20472: 20469: 20461: 20457: 20450: 20446: 20433: 20430: 20422: 20418: 20410: 20409: 20408: 20406: 20402: 20398: 20393: 20391: 20375: 20353: 20349: 20340: 20322: 20318: 20314: 20311: 20303: 20287: 20265: 20261: 20257: 20252: 20248: 20225: 20221: 20198: 20194: 20185: 20175: 20173: 20169: 20165: 20161: 20160:Oka principle 20156: 20152: 20148: 20138: 20136: 20118: 20097: 20080: 20050: 20032: 20013: 20001: 19998: 19973: 19942: 19927: 19907: 19888: 19873: 19871: 19846: 19793: 19785: 19773: 19756: 19744: 19727: 19715: 19700: 19692: 19691: 19690: 19688: 19683: 19667: 19634: 19612: 19581: 19559: 19547: 19544: 19539: 19527: 19524: 19504: 19501: 19498: 19478: 19475: 19472: 19464: 19449: 19427: 19405: 19393: 19390: 19385: 19356: 19336: 19329: 19313: 19291: 19274: 19247: 19245: 19202: 19190: 19187: 19176: 19171: 19157: 19137: 19114: 19106: 19084: 19072: 19069: 19064: 19047: 19035: 19032: 19027: 19011: 19010: 19009: 19008: 18992: 18972: 18964: 18946: 18888: 18876: 18873: 18863: 18859: 18844: 18842: 18838: 18837: 18832: 18831: 18811: 18808: 18805: 18779: 18773: 18767: 18760: 18757: 18753: 18750: 18747: 18743: 18740: 18736: 18733: 18707: 18701: 18694: 18691: 18687: 18683: 18680: 18673: 18667: 18664: 18661: 18650: 18632: 18626: 18619: 18616: 18612: 18609: 18606: 18603: 18600: 18573: 18570: 18566: 18563: 18537: 18531: 18525: 18522: 18519: 18508: 18506: 18492: 18472: 18449: 18446: 18443: 18417: 18400: 18398: 18388: 18386: 18353: 18335: 18320: 18316: 18300: 18294: 18274: 18268: 18248: 18242: 18222: 18216: 18210: 18198: 18195: 18192: 18189: 18186: 18182: 18178: 18175: 18172: 18169: 18166: 18163: 18161: 18145: 18135: 18132: 18125:For a domain 18118: 18062: 18047: 18043: 17993: 17985: 17982: 17978: 17957: 17954: 17951: 17931: 17925: 17922: 17919: 17911: 17907: 17891: 17888: 17885: 17877: 17873: 17869: 17853: 17847: 17844: 17830: 17814: 17799: 17795: 17779: 17776: 17770: 17764: 17750: 17748: 17732: 17729: 17723: 17717: 17694: 17691: 17688: 17679: 17676: 17670: 17664: 17641: 17638: 17630: 17622: 17616: 17613: 17610: 17602: 17598: 17594: 17589: 17585: 17578: 17572: 17566: 17558: 17539: 17536: 17528: 17520: 17514: 17511: 17508: 17500: 17496: 17492: 17487: 17483: 17476: 17470: 17464: 17421: 17411: 17407: 17392: 17376: 17373: 17365: 17340: 17337: 17334: 17331: 17328: 17325: 17322: 17319: 17311: 17303: 17277: 17274: 17271: 17263: 17259: 17255: 17250: 17246: 17242: 17239: 17231: 17216: 17214: 17198: 17178: 17170: 17154: 17123: 17119: 17103: 17089: 17081: 17036: 17013: 17008: 17004: 16993: 16990: 16987: 16983: 16979: 16976: 16969: 16968: 16967: 16966: 16962: 16917: 16914: 16909: 16905: 16896: 16890: 16887: 16886:Proposition 1 16883: 16867: 16850: 16831: 16828: 16817: 16813: 16804: 16800: 16785: 16781: 16768: 16764: 16752: 16746: 16741: 16728: 16723: 16720: 16717: 16714: 16711: 16707: 16703: 16697: 16691: 16684: 16669: 16666: 16661: 16657: 16648: 16644: 16632: 16626: 16615: 16610: 16607: 16604: 16600: 16596: 16593: 16587: 16581: 16571: 16570: 16569: 16555: 16535: 16529: 16526: 16518: 16499: 16496: 16493: 16487: 16484: 16458: 16455: 16452: 16446: 16443: 16421: 16382: 16372: 16369: 16361: 16343: 16315: 16300: 16297: 16274: 16254: 16248: 16237: 16234: 16226: 16222: 16218: 16215:be a complex 16214: 16204: 16187: 16167: 16161: 16150: 16147: 16139: 16136:be a complex 16135: 16116: 16096: 16090: 16079: 16076: 16068: 16064: 16060: 16041: 16035: 16029: 16023: 16020: 16017: 16014: 16011: 16000: 15984: 15977: 15974: 15970: 15966: 15948: 15936: 15933: 15919: 15917: 15899: 15882: 15872: 15870: 15854: 15834: 15831: 15821: 15803: 15785: 15766: 15756: 15743: 15739: 15730: 15725: 15714: 15709: 15706: 15702: 15698: 15690: 15687: 15683: 15676: 15671: 15667: 15659: 15643: 15621: 15592: 15572: 15569: 15565: 15554: 15545: 15537: 15532: 15521: 15517: 15514: 15503: 15484: 15478: 15475: 15472: 15463: 15444: 15412: 15406: 15392: 15389: 15386: 15383: 15376: 15375: 15374: 15373:the function 15360: 15351: 15348: 15338: 15334: 15314: 15308: 15297: 15294: 15291: 15284: 15282: 15266: 15258: 15254: 15253: 15238: 15232: 15229: 15226: 15223: 15220: 15217: 15214: 15206: 15203: 15193: 15192: 15173: 15170: 15167: 15164: 15158: 15152: 15145:the function 15144: 15143: 15126: 15116: 15113: 15110: 15107: 15085: 15075: 15064: 15061: 15058: 15055: 15052: 15049: 15046: 15036: 15035: 15034: 15032: 15028: 15007: 14995: 14992: 14985: 14981: 14967: 14958: 14952: 14939: 14936: 14933: 14926: 14923: 14922: 14916: 14914: 14910: 14904: 14902: 14881: 14877: 14869: 14865: 14862: 14859: 14834: 14830: 14823: 14820: 14810: 14806: 14780: 14777: 14772: 14768: 14745: 14710: 14706: 14680: 14674: 14666: 14647: 14637: 14634: 14611: 14603: 14593: 14588: 14584: 14558: 14554: 14550: 14547: 14531: 14527: 14506: 14502: 14498: 14493: 14483: 14468: 14460: 14456: 14452: 14449: 14433: 14429: 14413: 14410: 14407: 14398: 14384: 14381: 14378: 14372: 14364: 14346: 14336: 14312: 14292: 14289: 14286: 14277: 14275: 14271: 14253: 14243: 14236: 14233: 14225: 14209: 14200: 14198: 14179: 14161: 14142: 14138: 14134: 14128: 14115: 14112: 14096: 14090: 14080: 14077: 14074: 14066: 14055: 14049: 14041: 14038: 14035: 14032: 14028: 14024: 14019: 14009: 13998: 13997: 13996: 13994: 13990: 13974: 13971: 13968: 13960: 13941: 13923: 13907: 13887: 13865: 13855: 13852: 13823: 13820: 13817: 13809: 13805: 13788: 13784: 13780: 13775: 13771: 13748: 13744: 13721: 13717: 13708: 13705: 13687: 13683: 13672: 13669: 13666: 13662: 13658: 13655: 13635: 13632: 13627: 13623: 13619: 13614: 13610: 13601: 13584: 13580: 13574: 13569: 13566: 13563: 13559: 13555: 13552: 13530: 13526: 13522: 13519: 13516: 13511: 13507: 13498: 13497: 13491: 13489: 13485: 13481: 13460: 13457: 13437: 13434: 13431: 13422: 13420: 13404: 13401: 13398: 13390: 13386: 13382: 13378: 13362: 13359: 13356: 13353: 13350: 13330: 13327: 13324: 13302: 13292: 13289: 13269: 13266: 13263: 13255: 13237: 13222: 13218: 13213: 13196: 13193: 13190: 13179: 13169: 13166: 13158: 13154: 13150: 13146: 13142: 13138: 13135:, the domain 13122: 13111: 13093: 13083: 13080: 13072: 13065: 13040: 13031: 13029: 13025: 13021: 13020: 13016:'s notion of 13015: 12997: 12968: 12953: 12935: 12896: 12892: 12888: 12880: 12876: 12869: 12849: 12826: 12823: 12818: 12814: 12802: 12796: 12792: 12786: 12782: 12773: 12769: 12746: 12731: 12721: 12718: 12696: 12692: 12669: 12665: 12656: 12652: 12651: 12650: 12648: 12621: 12618: 12615: 12612: 12609: 12606: 12596: 12593: 12587: 12584: 12573: 12565: 12560: 12550: 12539: 12534: 12524: 12518: 12515: 12512: 12499: 12481: 12478: 12473: 12463: 12455: 12450: 12440: 12429: 12424: 12414: 12408: 12405: 12402: 12389: 12371: 12368: 12360: 12349: 12346: 12343: 12335: 12324: 12319: 12309: 12303: 12300: 12297: 12284: 12283: 12282: 12280: 12279:biholomorphic 12276: 12267: 12251: 12247: 12224: 12197: 12193: 12170: 12166: 12143: 12116: 12112: 12099: 12094: 12089: 12085: 12080: 12077: \ 12076: 12072: 12068: 12064: 12058: 12039: 12019: 12016: \ 12015: 12011: 12007: 12003: 11999: 11998: 11997: 11996: 11978: 11975: 11972: 11969: 11966: 11947: 11943: 11934: 11931: 11928: 11925: 11919: 11913: 11910: 11900: 11896: 11887: 11882: 11874: 11866: 11862: 11858: 11853: 11849: 11842: 11839: 11833: 11828: 11824: 11815: 11814: 11798: 11795: 11792: 11789: 11786: 11763: 11760: 11750: 11746: 11737: 11734: 11731: 11721: 11717: 11708: 11703: 11693: 11690: 11684: 11679: 11666: 11665: 11664: 11648: 11622: 11608: 11605: 11602: 11580: 11576: 11572: 11569: 11566: 11561: 11557: 11529: 11525: 11514: 11510: 11506: 11501: 11497: 11490: 11483: 11479: 11468: 11464: 11460: 11455: 11451: 11440: 11436: 11432: 11429: 11426: 11421: 11417: 11412: 11401: 11398: 11393: 11389: 11385: 11382: 11379: 11374: 11370: 11365: 11342: 11326: 11310: 11295: 11266: 11262: 11253: 11250: 11247: 11244: 11241: 11231: 11227: 11218: 11215: 11209: 11203: 11197: 11191: 11188: 11185: 11178: 11177: 11176: 11156: 11153: 11148: 11144: 11140: 11137: 11134: 11129: 11125: 11121: 11118: 11115: 11107: 11103: 11099: 11096: 11093: 11088: 11084: 11077: 11074: 11068: 11063: 11059: 11051: 11050: 11049: 11030: 11026: 11019: 11012:if the image 11011: 11007: 11002: 10994: 10992: 10988: 10984: 10980: 10976: 10972: 10953: 10949: 10945: 10942: 10939: 10936: 10933: 10930: 10927: 10921: 10917: 10911: 10907: 10903: 10898: 10893: 10889: 10884: 10880: 10876: 10870: 10866: 10862: 10857: 10853: 10848: 10844: 10836: 10832: 10828: 10825: 10822: 10817: 10813: 10806: 10803: 10799: 10791: 10790: 10789: 10775: 10772: 10767: 10763: 10754: 10750: 10731: 10727: 10723: 10720: 10717: 10714: 10711: 10708: 10702: 10697: 10694: 10690: 10685: 10681: 10678: 10673: 10669: 10664: 10660: 10657: 10654: 10651: 10648: 10640: 10636: 10632: 10629: 10626: 10621: 10617: 10610: 10607: 10603: 10595: 10594: 10593: 10579: 10576: 10571: 10567: 10557: 10542: 10533: 10529: 10525: 10520: 10515: 10511: 10500: 10496: 10492: 10488: 10483: 10475: 10469: 10465: 10461: 10456: 10452: 10447: 10438: 10419: 10416: 10413: 10410: 10407: 10404: 10401: 10392: 10388: 10378: 10376: 10357: 10353: 10349: 10346: 10343: 10340: 10337: 10334: 10331: 10325: 10321: 10315: 10311: 10307: 10302: 10297: 10293: 10288: 10284: 10280: 10274: 10270: 10266: 10261: 10257: 10252: 10248: 10240: 10236: 10232: 10229: 10226: 10221: 10217: 10210: 10207: 10203: 10195: 10194: 10193: 10179: 10176: 10168: 10163: 10159: 10155: 10152: 10149: 10144: 10139: 10135: 10128: 10123: 10119: 10096: 10086: 10078: 10074: 10070: 10067: 10064: 10059: 10055: 10048: 10045: 10025: 10022: 10019: 9997: 9987: 9984: 9975: 9966: 9964: 9960: 9956: 9940: 9937: 9934: 9913: 9887: 9874: 9871: 9848: 9835: 9832: 9824: 9808: 9800: 9796: 9780: 9769: 9765: 9761: 9757: 9753: 9749: 9731: 9721: 9718: 9713: 9703: 9683: 9680: 9677: 9669: 9653: 9630: 9627: 9624: 9621: 9615: 9612: 9606: 9603: 9595: 9576: 9543: 9530: 9527: 9504: 9491: 9488: 9466: 9452:be domain on 9451: 9441: 9439: 9435: 9431: 9427: 9423: 9407: 9404: 9401: 9387: 9385: 9381: 9377: 9373: 9357: 9337: 9331: 9328: 9323: 9320: 9316: 9295: 9289: 9286: 9283: 9261: 9246: 9242: 9238: 9233: 9225: 9223: 9219: 9215: 9210: 9196: 9193: 9188: 9184: 9161: 9132: 9128: 9124: 9116: 9112: 9108: 9103: 9099: 9092: 9072: 9069: 9061: 9057: 9053: 9048: 9044: 9037: 9029: 9025: 9021: 9003: 8993: 8990: 8982: 8957: 8951: 8948: 8945: 8939: 8933: 8927: 8922: 8915: 8911: 8905: 8899: 8891: 8890: 8889: 8887: 8867: 8861: 8858: 8855: 8849: 8846: 8840: 8834: 8819: 8815: 8811: 8805: 8802: 8799: 8793: 8787: 8781: 8776: 8769: 8765: 8759: 8753: 8745: 8729: 8725: 8721: 8718: 8713: 8704: 8698: 8695: 8692: 8689: 8684: 8680: 8676: 8673: 8670: 8667: 8662: 8658: 8654: 8651: 8646: 8637: 8631: 8622: 8613: 8607: 8602: 8593: 8582: 8579: 8576: 8573: 8570: 8566: 8557: 8554: 8544: 8541: 8538: 8529: 8519: 8511: 8508: 8505: 8497: 8491: 8488: 8485: 8476: 8470: 8467: 8464: 8458: 8451: 8450: 8449: 8448:, defined by 8432: 8429: 8426: 8423: 8420: 8397: 8389: 8370: 8367: 8364: 8358: 8336: 8321: 8305: 8297: 8293: 8289: 8288:wedge product 8273: 8253: 8228: 8213: 8209: 8206:Suppose that 8204: 8202: 8192: 8176: 8172: 8168: 8164: 8160: 8156: 8133: 8129: 8125: 8121: 8117: 8113: 8091: 8087: 8083: 8079: 8071: 8063: 8059: 8055: 8051: 8020: 8017: 8012: 8008: 8004: 8001: 7998: 7993: 7989: 7975: 7971: 7967: 7962: 7957: 7953: 7949: 7945: 7941: 7938: 7935: 7930: 7927: 7922: 7918: 7912: 7908: 7904: 7899: 7896: 7891: 7887: 7881: 7877: 7873: 7866: 7861: 7857: 7853: 7850: 7845: 7841: 7835: 7832: 7825: 7822: 7819: 7816: 7812: 7808: 7802: 7796: 7793: 7790: 7783: 7779: 7775: 7765: 7761: 7751: 7744: 7741: 7738: 7734: 7726: 7723: 7719: 7707: 7704: 7701: 7697: 7693: 7691: 7681: 7677: 7673: 7668: 7664: 7660: 7655: 7652: 7649: 7644: 7637: 7634: 7631: 7627: 7617: 7613: 7609: 7603: 7599: 7592: 7587: 7581: 7575: 7572: 7569: 7562: 7558: 7554: 7544: 7540: 7530: 7521: 7513: 7510: 7507: 7500: 7492: 7489: 7485: 7473: 7470: 7467: 7463: 7459: 7457: 7449: 7443: 7432: 7431: 7430: 7415: 7411: 7408: 7405: 7402: 7399: 7396: 7393: 7385: 7380: 7376: 7372: 7364: 7356: 7351: 7347: 7343: 7338: 7328: 7320: 7316: 7312: 7309: 7306: 7301: 7297: 7293: 7288: 7284: 7277: 7274: 7270: 7262: 7243: 7237: 7223: 7219: 7204: 7200: 7197: 7194: 7191: 7188: 7185: 7182: 7174: 7169: 7165: 7161: 7151: 7147: 7143: 7138: 7134: 7125: 7120: 7110: 7102: 7098: 7094: 7091: 7088: 7083: 7079: 7075: 7070: 7066: 7059: 7056: 7052: 7030: 7026: 7023: 7020: 7017: 7014: 7011: 7008: 7000: 6995: 6991: 6987: 6977: 6973: 6969: 6964: 6960: 6951: 6946: 6936: 6928: 6924: 6920: 6917: 6914: 6909: 6905: 6901: 6896: 6892: 6885: 6882: 6878: 6850: 6846: 6835: 6831: 6827: 6822: 6818: 6811: 6804: 6800: 6789: 6785: 6781: 6776: 6772: 6761: 6757: 6753: 6750: 6747: 6742: 6738: 6733: 6722: 6719: 6714: 6710: 6706: 6703: 6700: 6695: 6691: 6686: 6659: 6656: 6653: 6650: 6647: 6644: 6641: 6630: 6626: 6587: 6583: 6575: 6571: 6562: 6555: 6551: 6543: 6539: 6526: 6522: 6514: 6510: 6506: 6503: 6500: 6495: 6491: 6472: 6469: 6466: 6462: 6458: 6448: 6444: 6436: 6432: 6423: 6416: 6412: 6404: 6400: 6389: 6382: 6378: 6374: 6371: 6368: 6363: 6359: 6343: 6342: 6338: 6334: 6316: 6312: 6303: 6285: 6281: 6272: 6268: 6250: 6246: 6242: 6239: 6236: 6231: 6227: 6218: 6217: 6216: 6198: 6188: 6184: 6173: 6169: 6165: 6160: 6156: 6149: 6142: 6138: 6127: 6123: 6119: 6114: 6110: 6099: 6095: 6091: 6088: 6085: 6080: 6076: 6071: 6060: 6057: 6052: 6048: 6044: 6041: 6038: 6033: 6029: 6024: 6020: 6014: 6008: 6001: 6000: 5999: 5997: 5978: 5972: 5950: 5940: 5937: 5934: 5926: 5922: 5918: 5915: 5912: 5907: 5903: 5896: 5893: 5884: 5882: 5879:In addition, 5857: 5853: 5849: 5846: 5841: 5837: 5833: 5824: 5821: 5816: 5812: 5801: 5797: 5793: 5788: 5784: 5777: 5772: 5769: 5764: 5760: 5749: 5745: 5741: 5736: 5732: 5718: 5714: 5710: 5707: 5704: 5699: 5695: 5688: 5678: 5674: 5666: 5662: 5655: 5651: 5643: 5634: 5626: 5623: 5620: 5613: 5608: 5601: 5597: 5593: 5588: 5584: 5579: 5570: 5560: 5556: 5545: 5541: 5537: 5532: 5528: 5521: 5514: 5510: 5499: 5495: 5491: 5486: 5482: 5471: 5467: 5463: 5460: 5457: 5452: 5448: 5443: 5432: 5429: 5424: 5420: 5416: 5413: 5410: 5405: 5401: 5396: 5392: 5386: 5380: 5368: 5367: 5366: 5349: 5346: 5343: 5340: 5337: 5334: 5331: 5323: 5318: 5314: 5310: 5300: 5296: 5292: 5287: 5283: 5274: 5269: 5259: 5251: 5247: 5243: 5240: 5237: 5232: 5228: 5224: 5219: 5215: 5208: 5205: 5194: 5184: 5182: 5156: 5152: 5144: 5140: 5134: 5127: 5123: 5115: 5111: 5103: 5098: 5094: 5090: 5085: 5081: 5077: 5071: 5067: 5057: 5053: 5045: 5041: 5032: 5025: 5021: 5013: 5009: 4993: 4989: 4985: 4982: 4979: 4974: 4970: 4966: 4961: 4957: 4950: 4943: 4939: 4935: 4932: 4929: 4924: 4920: 4908: 4900: 4899: 4898: 4883: 4879: 4871: 4845: 4841: 4838: 4835: 4832: 4829: 4826: 4823: 4815: 4810: 4806: 4802: 4792: 4788: 4784: 4779: 4775: 4766: 4761: 4751: 4743: 4739: 4735: 4732: 4729: 4724: 4720: 4716: 4711: 4707: 4700: 4697: 4693: 4684: 4681:From (2), if 4679: 4646: 4638: 4631: 4629: 4615: 4610: 4606: 4602: 4599: 4594: 4590: 4586: 4577: 4574: 4569: 4565: 4554: 4550: 4546: 4541: 4537: 4530: 4525: 4522: 4517: 4513: 4502: 4498: 4494: 4489: 4485: 4471: 4467: 4463: 4460: 4457: 4452: 4448: 4441: 4431: 4427: 4419: 4415: 4408: 4404: 4396: 4387: 4379: 4376: 4373: 4365: 4360: 4356: 4352: 4347: 4343: 4336: 4326: 4322: 4314: 4310: 4301: 4294: 4290: 4282: 4278: 4262: 4258: 4254: 4251: 4248: 4243: 4239: 4235: 4230: 4226: 4219: 4212: 4208: 4204: 4201: 4198: 4193: 4189: 4173: 4172: 4169: 4167: 4166: 4151: 4144: 4142: 4126: 4122: 4118: 4115: 4110: 4106: 4102: 4090: 4086: 4082: 4077: 4073: 4066: 4058: 4054: 4050: 4045: 4041: 4027: 4023: 4019: 4016: 4013: 4008: 4004: 3997: 3987: 3983: 3975: 3971: 3964: 3960: 3952: 3943: 3935: 3932: 3929: 3922: 3917: 3909: 3905: 3901: 3898: 3895: 3890: 3886: 3879: 3872: 3871: 3868: 3867:. Therefore, 3866: 3862: 3858: 3842: 3810: 3806: 3802: 3790: 3786: 3782: 3777: 3773: 3766: 3758: 3754: 3750: 3745: 3741: 3729: 3725: 3721: 3716: 3712: 3698: 3694: 3690: 3687: 3684: 3679: 3675: 3671: 3666: 3662: 3655: 3645: 3641: 3633: 3627: 3623: 3619: 3611: 3607: 3599: 3595: 3590: 3586: 3582: 3574: 3570: 3562: 3553: 3545: 3542: 3539: 3532: 3527: 3525: 3515: 3511: 3507: 3495: 3491: 3487: 3482: 3478: 3466: 3462: 3458: 3453: 3449: 3435: 3431: 3427: 3424: 3421: 3416: 3412: 3408: 3403: 3399: 3395: 3390: 3386: 3379: 3369: 3365: 3357: 3351: 3347: 3343: 3335: 3331: 3323: 3314: 3306: 3303: 3300: 3293: 3288: 3286: 3276: 3272: 3268: 3259: 3255: 3251: 3246: 3242: 3231: 3227: 3223: 3220: 3217: 3212: 3208: 3204: 3199: 3195: 3188: 3178: 3174: 3166: 3159: 3156: 3153: 3149: 3144: 3142: 3132: 3128: 3124: 3121: 3118: 3113: 3109: 3102: 3091: 3090: 3089: 3087: 3069: 3064: 3061: 3058: 3050: 3026: 3022: 3019: 3016: 3013: 3010: 3007: 3004: 2994: 2990: 2986: 2982: 2976: 2972: 2968: 2963: 2959: 2954: 2950: 2945: 2935: 2927: 2923: 2919: 2916: 2913: 2908: 2904: 2900: 2895: 2891: 2884: 2881: 2877: 2873: 2867: 2864: 2861: 2825: 2821: 2817: 2814: 2811: 2806: 2802: 2798: 2793: 2789: 2784: 2740: 2724: 2721: 2710: 2706: 2702: 2699: 2696: 2691: 2687: 2683: 2678: 2674: 2642: 2638: 2634: 2631: 2628: 2623: 2619: 2615: 2610: 2606: 2580: 2576: 2553: 2549: 2528: 2525: 2522: 2519: 2516: 2513: 2510: 2507: 2504: 2482: 2465: 2462:is piecewise 2447: 2443: 2422: 2415: 2411: 2407: 2397: 2384: 2381: 2378: 2358: 2342: 2339: 2336: 2323: 2319: 2308: 2298: 2292: 2289: 2286: 2283: 2280: 2274: 2271: 2260: 2255: 2237: 2233: 2224: 2215: 2212: 2204: 2200: 2191: 2170: 2166: 2157: 2148: 2140: 2136: 2127: 2117: 2111: 2108: 2105: 2102: 2099: 2093: 2090: 2067: 2047: 2027: 1997: 1987: 1984: 1975: 1959: 1950: 1942: 1933: 1930: 1924: 1915: 1907: 1888: 1879: 1871: 1862: 1856: 1847: 1839: 1813: 1805: 1789: 1769: 1744: 1741: 1708: 1705: 1691: 1689: 1685: 1681: 1677: 1673: 1669: 1665: 1660: 1642: 1638: 1634: 1631: 1628: 1623: 1620: 1617: 1613: 1609: 1606: 1603: 1598: 1595: 1592: 1588: 1584: 1581: 1578: 1573: 1569: 1562: 1556: 1548: 1546: 1541: 1539: 1535: 1532:The function 1530: 1511: 1502: 1499: 1493: 1487: 1484: 1478: 1472: 1469: 1463: 1460: 1457: 1451: 1443: 1419: 1409: 1406: 1386: 1383: 1380: 1336: 1326: 1323: 1315: 1300: 1284: 1255: 1240: 1236: 1232: 1228: 1224: 1205: 1200: 1190: 1182: 1177: 1173: 1169: 1164: 1160: 1152: 1151: 1150: 1149: 1130: 1125: 1119: 1114: 1107: 1104: 1099: 1093: 1084: 1083: 1082: 1081: 1076: 1072: 1068: 1063: 1059: 1055: 1054:complex plane 1051: 1046: 1041: 1037: 1032: 1031: 1025: 1017: 1013: 1008: 1005: 999: 981: 978: 964: 946: 931: 904: 898: 894: 887: 883: 865: 850: 832: 803: 765: 760: 742: 728: 718: 716: 712: 708: 704: 703:hyperfunction 699: 697: 693: 687: 682: 678: 674: 670: 666: 662: 661:modular forms 658: 657:number theory 654: 650: 646: 642: 638: 636: 632: 628: 624: 623: 618: 617:D. C. Spencer 614: 610: 606: 602: 598: 594: 589: 587: 582: 578: 574: 573: 568: 567: 548: 533: 527: 500: 496: 492: 488: 484: 479: 477: 473: 469: 463: 458: 452: 425: 415: 412: 404: 400: 396: 392: 388: 384: 383:Peter Thullen 380: 376: 372: 366: 361: 360:Pierre Cousin 357: 354:With work of 352: 350: 346: 345:branch points 342: 338: 334: 330: 326: 322: 318: 314: 310: 300: 298: 294: 276: 247: 229: 225: 209: 206: 203: 197: 192: 182: 179: 171: 150: 147: 139: 135: 132:-dimensional 127: 124:; or locally 123: 119: 108: 104: 100: 99: 92: 87: 82: 80: 77:), which the 76: 72: 68: 46: 32: 28: 24: 19: 31330:Pseudoconvex 31295: 31263: 31215: 31189: 31185: 31162: 31158: 31135: 31131: 31109: 31084: 31080: 31056: 31052: 30997: 30971: 30953: 30935: 30917: 30899: 30881: 30863: 30845: 30827: 30809: 30791: 30773: 30755: 30737: 30719: 30701: 30668: 30632: 30605: 30579: 30553: 30525: 30504: 30483: 30463: 30430: 30414: 30388: 30361: 30333: 30307: 30287: 30266: 30249: 30230: 30208: 30181: 30143: 30139: 30129: 30110: 30106: 30096: 30069: 30065: 30055: 30030: 30026: 30020: 29995: 29991: 29985: 29963:(1): 28–48. 29960: 29956: 29950: 29923: 29919: 29909: 29882: 29876: 29849: 29843: 29818: 29814: 29804: 29777: 29771: 29754: 29750: 29740: 29708: 29704: 29691: 29638: 29634: 29624: 29599: 29595: 29589: 29564: 29560: 29554: 29498: 29488: 29460: 29453: 29437:. Springer. 29433: 29426: 29398: 29391: 29366: 29362: 29356: 29337: 29333: 29323: 29309: 29299: 29272: 29266: 29256: 29237: 29233: 29223: 29206: 29202: 29192: 29167: 29163: 29157: 29132: 29128: 29122: 29095: 29091: 29081: 29053: 29046: 29021: 29017: 29007: 28954: 28950: 28944: 28919: 28915: 28909: 28877: 28873: 28835: 28831: 28825: 28790: 28786: 28760: 28756: 28750: 28731: 28727: 28717: 28698: 28694: 28684: 28659: 28655: 28645: 28618: 28614: 28601: 28569: 28565: 28559: 28524: 28520: 28514: 28465: 28459: 28434: 28430: 28424: 28392: 28388: 28346: 28324: 28320: 28310: 28291: 28287: 28277: 28258: 28254: 28244: 28225: 28215: 28196: 28192: 28158: 28154: 28148: 28140: 28113: 28107: 28089: 28069: 28062: 28042: 28035: 28001: 27997: 27984: 27956: 27949: 27921: 27914: 27881: 27877: 27867: 27835: 27831: 27774: 27767: 27731: 27724: 27692:(2): 19–24. 27689: 27685: 27659: 27655: 27616: 27599: 27595: 27589: 27557: 27553: 27547: 27523: 27519: 27513: 27479: 27475: 27469: 27455: 27445: 27413: 27409: 27399: 27380: 27376: 27366: 27339: 27333: 27323: 27298: 27294: 27284: 27267: 27263: 27235: 27231: 27199: 27185: 27175: 27156: 27152: 27113: 27109: 27077: 27073: 27063: 27044: 27040: 27030: 27010: 27003: 26987:. Springer. 26983: 26976: 26939: 26935: 26925: 26906: 26902: 26892: 26867: 26863: 26857: 26830: 26809:the original 26778: 26768: 26740: 26734: 26701: 26697: 26673: 26663: 26638: 26610: 26606: 26571: 26567: 26550: 26546: 26528: 26503: 26499: 26474:Noguchi, J. 26469: 26451: 26416: 26412: 26373: 26369: 26363: 26337: 26333: 26298: 26292: 26260: 26256: 26246: 26219: 26215: 26205: 26178: 26174: 26164: 26147: 26141: 26135: 26116: 26112: 26102: 26075:math/0610985 26065: 26061: 26051: 26024: 26018: 26008: 25968: 25964: 25923: 25919: 25913: 25885: 25878: 25851: 25847: 25837: 25809: 25802: 25774: 25763: 25736: 25732: 25722: 25695: 25669: 25665: 25659: 25645: 25635: 25616: 25612: 25602: 25559: 25553: 25508: 25502: 25477: 25473: 25419: 25415: 25369: 25365: 25311: 25307: 25274: 25267: 25240: 25236: 25118: 25108: 25091: 25010: 25001: 24992: 24983: 24974: 24965: 24956: 24944: 24935: 24925: 24916: 24907: 24865: 24848: 24839: 24830: 24817: 24807: 24772: 24767: 24754: 24745: 24728: 24709: 24613:. Also, the 24611:Hodge theory 24605: 24592: 24584: 24580: 24525: 24518:Grothendieck 24408: 24395: 24384: 24369: 24360: 24250: 24190: 24186: 24182: 24178: 24116:Stein domain 24115: 24111: 24084:Levi problem 24083: 24079: 24075: 23970: 23942: 23932: 23928: 23924: 23923:, there are 23894: 23872: 23868: 23858: 23807: 23803: 23796: 23592:, such that 23518: 23435: 23431: 23429: 23426: 23418: 23412: 23357: 23351: 23345: 23341: 23305: 23301: 23295: 23289: 23175: 23169: 23157: 23151: 23147: 23143: 23139: 23135: 23101: 23097: 23095: 23063: 23057: 23053: 23049: 23045: 23041: 23006: 22994: 22942:, therefore 22839: 22597: 22590:Helmut Röhrl 22586:Hans Grauert 22583: 22578: 22575:deep theorem 22566: 22564: 22556: 22425: 22391: 22385: 22381: 22222: 22192: 22185: 22181: 22177: 22113: 22111: 22055: 22051: 22048:vector space 22039: 22037: 22003: 21999: 21937: 21933: 21864: 21856: 21830: 21824: 21683: 21681: 21476: 21443: 21240: 21181: 21024: 20773: 20733: 20669: 20659: 20651: 20647: 20643: 20641: 20508: 20404: 20400: 20396: 20394: 20338: 20301: 20183: 20181: 20171: 20167: 20163: 20144: 20101: 19869: 19684: 19650: 19272: 19174: 19172: 19129: 18862:ringed space 18857: 18855: 18834: 18828: 18794: 18505:, such that 18406: 18394: 18315:Levi problem 18202: 18196: 18190: 18184: 18176: 18170: 18164: 18124: 18045: 18041: 17912:. i.e., let 17909: 17905: 17875: 17871: 17867: 17836: 17793: 17756: 17746: 17556: 17390: 17229: 17227: 17212: 17168: 17117: 17095: 17087: 17028: 16964: 16892: 16888: 16885: 16884: 16848: 16846: 16516: 16359: 16326: 16224: 16220: 16216: 16212: 16210: 16137: 16133: 16066: 16062: 15998: 15969:pseudoconvex 15968: 15964: 15925: 15915: 15880: 15878: 15819: 15786: 15605:is of class 15501: 15464: 15433: 15341: 15256: 15026: 15024: 14983: 14906: 14900: 14708: 14704: 14678: 14676: 14672: 14529: 14525: 14431: 14427: 14399: 14362: 14278: 14273: 14269: 14223: 14201: 14196: 14159: 14157: 13992: 13988: 13958: 13920:dimensional 13844: 13480:accumulating 13423: 13418: 13388: 13384: 13380: 13376: 13253: 13252:is called a 13220: 13216: 13214: 13156: 13152: 13148: 13144: 13140: 13136: 13109: 13070: 13068: 13063: 13027: 13018: 12951: 12919: 12654: 12645: 12497:(unit ball); 12273: 12103: 12087: 12083: 12078: 12074: 12066: 12062: 12056: 12017: 12013: 12001: 11633: 11625:Some results 11327: 11291: 11174: 11005: 11003: 11000: 10978: 10970: 10968: 10752: 10748: 10746: 10558: 10436: 10379: 10374: 10372: 9976: 9972: 9962: 9958: 9954: 9822: 9798: 9794: 9767: 9763: 9759: 9755: 9751: 9747: 9593: 9481:, such that 9449: 9447: 9421: 9393: 9383: 9379: 9375: 9371: 9370:is called a 9244: 9240: 9236: 9234: 9231: 9211: 9027: 9023: 9019: 8978: 8885: 8883: 8410:of bidegree 8319: 8295: 8291: 8211: 8207: 8205: 8198: 8042: 7229: 7220: 6614: 6336: 6332: 6301: 6270: 6266: 6213: 5995: 5885: 5880: 5878: 5192: 5191:If function 5190: 5178: 4682: 4680: 4644: 4643: 4632: 4163: 4161: 4145: 3856: 3831: 2409: 2405: 2403: 2359:, as : 2256: 1976: 1697: 1683: 1671: 1667: 1663: 1661: 1549: 1544: 1542: 1537: 1533: 1531: 1444: 1313: 1311: 1220: 1145: 1074: 1070: 1066: 1047: 1029: 1023: 1009: 1003: 902: 763: 724: 700: 648: 644: 641:C. L. Siegel 639: 634: 630: 620: 590: 570: 564: 531: 525: 498: 487:Hans Grauert 483:Henri Cartan 480: 461: 450: 353: 341:ramification 306: 107:power series 102: 96: 90: 83: 70: 22: 20: 18: 29821:: 129–133. 29757:: 128–130. 29209:: 118–121. 28734:: 223–245. 28662:: 233–259. 28572:: 201–222, 28395:: 430–461, 28294:: 115–130. 28261:: 245–255. 27560:: 152–183, 27383:: 153–164. 27342:: 243–254. 27301:: 119–123. 27276:2433/263965 27047:: 427–474. 26870:: 204–216. 26704:(1): 1–73. 26263:: 617–647. 26181:: 244–259. 25971:: 223–242, 25819:10.4171/049 25739:: 185–220. 25480:(4): 8–13. 24950:Oka's lemma 24900:Oka's lemma 24898:domain.See 24802:is compact. 24713:That is an 24670:CR manifold 24355:is a Stein 22844:shows that 22118:paracompact 22056:Stein space 22044:submanifold 20388:shares the 20339:holomorphic 19273:finite type 19220:is a sheaf 18906:is a sheaf 17232:-functions 16965:, such that 16930:with class 14760:defined by 14202:The domain 13014:Kiyoshi Oka 12387:(polydisc); 12239:containing 11048:of the set 9961:other than 5179:Therefore, 4678:-function. 1312:A function 1148:determinant 1021:(such that 789:, and when 375:Kiyoshi Oka 363: [ 315:, and some 122:polynomials 98:holomorphic 65:-tuples of 61:, that is, 27:mathematics 31345:Categories 31298:PlanetMath 31224:1044.01520 31044:0367.14001 29926:: 97–136. 29763:0050.17701 29546:0367.14001 29215:0070.30401 28902:0108.07804 28800:1610.07768 28769:0192.18304 28566:Math. Ann. 28479:1802.03924 28417:0038.23502 28011:1704.07726 27699:1807.08246 27582:0073.30301 27438:0043.30301 27132:0057.31503 26787:0053.05302 26674:numdam.org 26655:0053.05301 26340:: 97–155, 26315:0060.24006 26156:0001.28501 25999:1608.00950 25977:37.0443.01 25594:0075.30401 25454:0974.32001 25243:: 89–152. 25181:References 24858:Jean Leray 24701:Annotation 24570:positivity 24514:Hirzebruch 24512:in 1938. 24088:E. E. Levi 23853:proper map 23423:K-complete 23352:and also, 23100:manifolds 22467:such that 22398:, i.e. if 22384:subset of 22380:is also a 22180:. We call 22108:Definition 20736:such that 20304:such that 19874:the sheaf 19682:-modules. 19491:, integer 18852:Definition 18319:E. E. Levi 17798:unramified 17753:Definition 16061:subset of 15973:continuous 15914:, we call 15818:-function 15025:is called 14924:A function 14222:is called 14162:by taking 13806:The first 12071:complement 11294:convex set 11008:is called 9903:such that 9438:Lie groups 8298:then, for 2464:smoothness 1688:continuous 1442:such that 1308:Definition 767:copies of 690:, and the 387:Karl Stein 371:E. E. Levi 168:), is the 31036:197660097 30978:EMS Press 30960:EMS Press 30942:EMS Press 30924:EMS Press 30906:EMS Press 30888:EMS Press 30870:EMS Press 30852:EMS Press 30834:EMS Press 30816:EMS Press 30798:EMS Press 30780:EMS Press 30762:EMS Press 30744:EMS Press 30726:EMS Press 30708:EMS Press 30659:125752012 30461:(1990) , 30258:896179082 30173:Textbooks 30162:122894640 30088:121799985 29835:116472982 29733:123643759 29538:197660097 29316:EMS Press 28998:0905.2343 28971:179177434 28936:120565581 28852:122214512 28817:119670805 28763:: 29–35. 28701:: 77–99. 28676:122840967 28628:1108.2078 28594:122647212 28551:119631664 28534:1010.3738 28506:220266044 28451:122162708 28409:122535410 28344:(2009) , 28028:119697608 27906:121855488 27802:125752012 27759:125752012 27716:119619733 27574:117913713 27506:122862268 27462:EMS Press 27430:120455177 27358:123827662 27315:121224216 27270:: 27–46. 27192:EMS Press 27159:: 37–48. 27080:: 45–80. 26884:123982856 26781:: 67–58. 26641:: 41–55. 26613:: 29–64. 26443:119685542 26426:1303.6933 26419:: 21–45. 26392:119837287 26376:: 63–91, 26356:0075-3432 26307:0040-8735 26301:: 15–52, 26238:124324696 26197:121072397 26150:: 1–116. 25940:121700550 25870:118248529 25755:123480258 25652:EMS Press 25619:: 21–38. 25578:0373-0956 25494:121138963 25259:120051843 25233:operator" 25218:¯ 25215:∂ 25113:manifold. 25033:× 24788:ν 24718:connected 24522:morphisms 24465:≅ 24459:^ 24334:∞ 24331:− 24320:− 24307:That is, 24283:∞ 24280:− 24269:− 24219:− 24156:≤ 24144:ψ 24141:≤ 24138:∞ 24135:− 24132:∣ 24098:ψ 24056:≤ 24044:ψ 24041:∣ 24035:∈ 24003:ψ 23997:¯ 23994:∂ 23988:∂ 23957:ψ 23908:∈ 23698:… 23653:− 23639:∈ 23492:… 23456:∈ 23382:∈ 23375:⋃ 23325:→ 23273:∩ 23218:∂ 23215:∈ 23119:⊂ 23025:⊂ 22789:⟶ 22759:⟶ 22751:∗ 22715:⟶ 22641:∗ 22487:≠ 22439:∈ 22409:≠ 22342:∈ 22336:∀ 22300:∈ 22289:≤ 22258:∈ 22241:¯ 22206:⊂ 21857:immersion 21637:→ 21604:π 21598:→ 21590:∗ 21561:→ 21455:π 21426:→ 21421:∗ 21396:→ 21385:π 21379:→ 21307:∗ 21275:∗ 21216:∗ 21162:∗ 21133:→ 21125:∗ 21108:∗ 21076:ϕ 21062:∗ 21002:∗ 20985:∗ 20953:ϕ 20939:∗ 20890:ϕ 20868:∗ 20851:∗ 20818:∗ 20789:∗ 20600:→ 20557:ϕ 20451:ϕ 20395:Now, let 20315:− 20258:− 20054:⟩ 20048:⟨ 19791:→ 19762:→ 19733:→ 19704:→ 19635:φ 19565:→ 19545:⊕ 19525:φ 19476:⊆ 19411:→ 19391:⊕ 19112:→ 19090:→ 19070:⊕ 19053:→ 19033:⊕ 18812:δ 18768:∈ 18758:δ 18754:∩ 18751:δ 18702:∈ 18692:δ 18668:δ 18651:For each 18627:∈ 18617:δ 18613:∩ 18610:δ 18571:δ 18532:∈ 18526:δ 18493:δ 18450:δ 18433:of pairs 18385:L methods 18368:¯ 18365:∂ 18298:⇒ 18272:⇒ 18246:⇒ 18220:⇔ 18214:⇔ 18136:⊂ 18105:Ω 18085:Ω 18026:∞ 18009:admits a 17983:− 17979:ψ 17955:∈ 17929:→ 17920:ψ 17889:∩ 17851:∂ 17848:∈ 17777:⊂ 17730:⊂ 17677:⊂ 17599:φ 17537:≤ 17497:φ 17445:Δ 17419:∂ 17408:φ 17404:∂ 17338:≤ 17332:≤ 17320:≤ 17301:Δ 17260:φ 17240:φ 17199:ρ 17179:ρ 17155:ρ 17104:ρ 17066:∞ 17037:φ 16999:∞ 16984:⋃ 16947:∞ 16915:⊂ 16829:≥ 16823:¯ 16791:¯ 16775:∂ 16761:∂ 16747:ρ 16738:∂ 16708:∑ 16698:ρ 16682:, we have 16641:∂ 16627:ρ 16624:∂ 16601:∑ 16582:ρ 16579:∇ 16533:∂ 16530:∈ 16494:ρ 16482:∂ 16453:ρ 16405:which is 16388:→ 16370:ρ 16318:Levi form 16301:ψ 16267:∞ 16255:∩ 16238:∈ 16235:ψ 16180:∞ 16168:∩ 16151:∈ 16148:ψ 16109:∞ 16097:∩ 16080:∈ 16077:ψ 16036:≤ 16024:φ 16015:∈ 15985:φ 15937:⊂ 15849:¯ 15846:∂ 15840:∂ 15832:− 15760:¯ 15750:∂ 15736:∂ 15722:∂ 15703:λ 15684:λ 15570:≥ 15558:¯ 15550:∂ 15543:∂ 15529:∂ 15512:Δ 15445:⊂ 15442:Δ 15416:∞ 15413:− 15407:∪ 15399:→ 15396:Δ 15393:: 15390:φ 15387:∘ 15358:→ 15355:Δ 15352:: 15349:φ 15318:∞ 15315:− 15309:∪ 15301:→ 15295:: 15230:∈ 15207:∈ 15156:↦ 15117:∈ 15076:⊂ 15065:∈ 15029:if it is 14996:⊂ 14962:∞ 14959:− 14953:∪ 14943:→ 14937:: 14866:⁡ 14860:− 14778:∈ 14719:Ω 14638:⊂ 14609:∖ 14487:^ 14411:≥ 14382:⊂ 14376:∖ 14340:^ 14247:^ 14116:∈ 14078:∈ 14067:≤ 14036:∈ 14013:^ 13972:⊂ 13856:⊂ 13821:≥ 13781:× 13678:∞ 13663:⋃ 13636:⋯ 13633:⊆ 13620:⊆ 13585:ν 13564:ν 13560:⋂ 13520:… 13461:⊂ 13360:∩ 13354:⊂ 13293:⊂ 13267:⊂ 13194:≥ 13170:⊂ 13120:∂ 13084:⊂ 13050:Ω 12870:φ 12850:σ 12797:σ 12779:↦ 12761:given by 12737:→ 12719:φ 12619:≠ 12525:∈ 12415:∈ 12310:∈ 12252:ε 12221:Δ 12198:ε 12171:ε 12140:Δ 12117:ε 12069:. If the 11973:ε 11932:ε 11929:− 11920:∪ 11914:ε 11879:Δ 11875:∈ 11829:ε 11793:ε 11694:∈ 11676:Δ 11570:… 11507:− 11491:⋯ 11461:− 11430:… 11407:∞ 11383:… 11366:∑ 11254:⁡ 11245:… 11219:⁡ 11198:λ 11195:→ 11186:λ 11154:≠ 11138:… 11116:∈ 11097:… 11064:∗ 11031:∗ 11020:λ 10975:star-like 10940:… 10928:ν 10912:ν 10904:− 10894:ν 10881:≤ 10871:ν 10863:− 10858:ν 10826:… 10773:∈ 10724:π 10715:θ 10712:≤ 10698:θ 10679:− 10630:… 10577:∈ 10534:ν 10526:− 10516:ν 10501:ν 10497:θ 10480:→ 10470:ν 10462:− 10414:… 10402:ν 10393:ν 10389:θ 10373:A domain 10344:… 10332:ν 10316:ν 10308:− 10298:ν 10275:ν 10267:− 10262:ν 10230:… 10177:∈ 10153:… 10087:∈ 10068:… 10023:≥ 9988:⊂ 9938:≥ 9911:∂ 9875:∈ 9836:∈ 9778:∂ 9681:∩ 9634:∅ 9631:≠ 9625:∩ 9531:∈ 9492:∈ 9358:ϕ 9335:→ 9321:− 9317:ϕ 9293:→ 9284:ϕ 8994:⊂ 8946:ζ 8940:ω 8934:ζ 8920:∂ 8916:∫ 8856:ζ 8850:ω 8847:∧ 8841:ζ 8830:¯ 8827:∂ 8816:∫ 8812:− 8800:ζ 8794:ω 8788:ζ 8774:∂ 8770:∫ 8726:ζ 8719:∧ 8708:¯ 8705:ζ 8696:∧ 8693:⋯ 8690:∧ 8681:ζ 8674:∧ 8671:⋯ 8668:∧ 8659:ζ 8652:∧ 8641:¯ 8638:ζ 8617:¯ 8608:− 8597:¯ 8594:ζ 8580:≤ 8574:≤ 8567:∑ 8545:ζ 8542:− 8509:π 8489:− 8465:ζ 8459:ω 8430:− 8398:ζ 8365:ζ 8359:ω 8306:ζ 8274:∧ 8251:∂ 8177:ν 8161:ν 8134:ν 8118:ν 8088:ν 8056:ν 8009:α 8002:⋯ 7990:α 7963:⋅ 7958:ζ 7939:⋯ 7928:− 7919:α 7909:ζ 7905:⋯ 7897:− 7888:α 7878:ζ 7874:⋅ 7858:α 7854:⋯ 7842:α 7823:⋯ 7809:× 7803:ζ 7797:ω 7794:∫ 7791:⋯ 7784:ν 7766:ν 7762:ζ 7752:∫ 7742:π 7713:∞ 7698:∑ 7674:⋅ 7669:ζ 7635:− 7632:ζ 7588:× 7582:ζ 7576:ω 7573:∫ 7570:⋯ 7563:ν 7545:ν 7541:ζ 7531:∫ 7511:π 7479:∞ 7464:∑ 7444:ω 7406:… 7394:ν 7381:ν 7352:ν 7329:∈ 7310:… 7238:ω 7195:… 7183:ν 7170:ν 7152:ν 7144:− 7139:ν 7111:∈ 7092:… 7021:… 7009:ν 6996:ν 6978:ν 6970:− 6965:ν 6937:∈ 6918:… 6828:− 6812:⋯ 6782:− 6751:… 6728:∞ 6704:… 6687:∑ 6654:… 6642:ν 6631:ν 6566:∂ 6563:⋯ 6534:∂ 6504:⋯ 6487:∂ 6478:∞ 6463:∑ 6427:∂ 6424:⋯ 6395:∂ 6372:⋯ 6355:∂ 6240:… 6166:− 6150:⋯ 6120:− 6089:… 6066:∞ 6042:… 6025:∑ 5941:⊂ 5935:∈ 5916:… 5854:ζ 5847:⋯ 5838:ζ 5794:− 5785:ζ 5778:⋯ 5742:− 5733:ζ 5715:ζ 5708:… 5696:ζ 5671:∂ 5667:∫ 5663:⋯ 5648:∂ 5644:∫ 5624:π 5594:⋯ 5538:− 5522:⋯ 5492:− 5461:… 5438:∞ 5414:… 5397:∑ 5344:… 5332:ν 5319:ν 5301:ν 5293:− 5288:ν 5260:∈ 5241:… 5135:⋯ 5091:⋯ 5072:≤ 5036:∂ 5033:⋯ 5006:∂ 4990:ζ 4983:… 4971:ζ 4958:ζ 4933:⋯ 4916:∂ 4880:≤ 4836:… 4824:ν 4811:ν 4803:≤ 4793:ν 4785:− 4780:ν 4776:ζ 4752:∈ 4740:ζ 4733:… 4721:ζ 4708:ζ 4698:ζ 4664:∞ 4647:is class 4607:ζ 4600:⋯ 4591:ζ 4547:− 4538:ζ 4531:⋯ 4495:− 4486:ζ 4468:ζ 4461:… 4449:ζ 4424:∂ 4420:∫ 4416:⋯ 4401:∂ 4397:∫ 4377:π 4353:⋯ 4305:∂ 4302:⋯ 4273:∂ 4259:ζ 4252:… 4240:ζ 4227:ζ 4202:⋯ 4185:∂ 4168:) we get 4123:ζ 4116:⋯ 4107:ζ 4083:− 4074:ζ 4067:⋯ 4051:− 4042:ζ 4024:ζ 4017:… 4005:ζ 3980:∂ 3976:∫ 3972:⋯ 3957:∂ 3953:∫ 3933:π 3899:… 3840:∂ 3807:ζ 3783:− 3774:ζ 3767:⋯ 3751:− 3742:ζ 3722:− 3713:ζ 3695:ζ 3688:… 3676:ζ 3663:ζ 3638:∂ 3634:∫ 3624:ζ 3604:∂ 3600:∫ 3596:⋯ 3587:ζ 3567:∂ 3563:∫ 3543:π 3512:ζ 3488:− 3479:ζ 3459:− 3450:ζ 3425:… 3400:ζ 3387:ζ 3362:∂ 3358:∫ 3348:ζ 3328:∂ 3324:∫ 3304:π 3273:ζ 3252:− 3243:ζ 3221:… 3196:ζ 3171:∂ 3167:∫ 3157:π 3122:… 3059:ν 3017:… 3005:ν 2995:ν 2987:≤ 2977:ν 2969:− 2964:ν 2960:ζ 2936:∈ 2924:ζ 2917:… 2905:ζ 2892:ζ 2882:ζ 2854:¯ 2851:Δ 2818:× 2815:⋯ 2812:× 2799:× 2785:⊂ 2780:¯ 2777:Δ 2753:¯ 2750:Δ 2722:∈ 2717:¯ 2703:× 2700:⋯ 2697:× 2684:× 2649:¯ 2635:× 2632:⋯ 2629:× 2616:× 2581:ν 2577:γ 2554:ν 2523:… 2505:ν 2448:ν 2444:γ 2423:γ 2373:¯ 2370:∂ 2329:¯ 2314:∂ 2306:∂ 2287:… 2275:∈ 2269:∀ 2230:∂ 2222:∂ 2216:− 2197:∂ 2189:∂ 2163:∂ 2155:∂ 2133:∂ 2125:∂ 2106:… 2094:∈ 2088:∀ 2003:→ 1948:∂ 1940:∂ 1934:− 1913:∂ 1905:∂ 1877:∂ 1869:∂ 1845:∂ 1837:∂ 1745:∈ 1717:→ 1632:… 1596:− 1582:… 1560:↦ 1515:‖ 1509:‖ 1425:→ 1384:∈ 1327:⊂ 1235:Jacobians 1105:− 1052:. On the 895:over the 649:functions 448:whenever 431:→ 403:removable 373:, and of 325:parameter 207:≥ 183:⊂ 151:⊂ 31318:polydisc 31290:BY-NC-SA 30996:(1977). 30608:. 2005. 30423:14003250 30331:(2011). 30229:(1981). 29711:: 9–26, 29699:(1955), 29683:16589138 29496:(1977). 29038:43736735 28199:: 7–19. 28161:: 1–61. 27823:(1955), 27608:43698735 27482:: 1–22, 27416:: 1–16, 26942:(2): 1. 26726:73633995 26718:41342411 26506:: 1–27. 25959:(1906), 25854:: 1–12. 25678:43700400 25562:: 1–42. 25548:(1956). 25438:43833225 25330:53582451 25099:, as in 24654:See also 23262:so that 22143:and let 22112:Suppose 21869:embedded 21405:→ 21072:→ 20949:→ 20553:→ 20447:→ 20390:singular 18830:coherent 18761:′ 18744:′ 18695:′ 18684:′ 18620:′ 18574:′ 17147:passing 16436:so that 16290:, i.e., 16225:strictly 14532:, where 13484:boundary 13424:For the 13328:⊄ 12021:, where 9430:Poincaré 8165:′ 8122:′ 8105:, where 8092:′ 8060:′ 5998: : 3832:Because 3040:and let 2739:polydisc 2466:, class 1826: : 1678:, or as 1050:oriented 1000:is thus 996:and its 647:had few 468:manifold 351:theory. 31101:2316199 31073:2323391 31028:0463157 30407:0580152 30047:1969750 30012:2372375 29977:1969701 29942:0116022 29725:0067489 29674:1063603 29643:Bibcode 29616:1969802 29581:2372120 29530:0463157 29383:2007052 29291:2047046 29184:2946547 29149:2372949 29114:1994247 28894:1970257 28637:3086750 28586:0043219 28370:0069903 28327:: 1–26. 27898:0217083 27860:0068874 27852:1969915 27540:1969189 26647:0064154 26043:2035718 25586:0082175 25446:1799753 25388:0477104 24566:Kodaira 24560:. The 24399:fibrant 24385:In the 24357:filling 23977:) with 23146:, then 23098:complex 23011:If the 22520:to the 22382:compact 22046:of the 20509:By the 19326:has an 18963:modules 18723:, then 18397:sheaves 18391:Sheaves 18317:(after 16895:bounded 16515:. Now, 15636:, then 13219:in the 12275:Thullen 9243:of the 7261:annulus 1038:by the 757:is the 679:from a 31270: 31222: 31116: 31099: 31071: 31042: 31034: 31026: 31016: 30676: 30657: 30647: 30620: 30594: 30568: 30541: 30512: 30491: 30471: 30445: 30421: 30405: 30395: 30376: 30349: 30315: 30294: 30273: 30256: 30237: 30215: 30196: 30160: 30086: 30045: 30010: 29975: 29940: 29897: 29864: 29833: 29792: 29761: 29731: 29723: 29681: 29671: 29663: 29614: 29579: 29544: 29536: 29528: 29518: 29476: 29441: 29414: 29381: 29289: 29213: 29182: 29147: 29112: 29069: 29036: 28969: 28934: 28900: 28892: 28850: 28815: 28767: 28674: 28635: 28592: 28584: 28549: 28504: 28494: 28449: 28415: 28407: 28368: 28358: 28232: 28128: 28077: 28050: 28026: 27972: 27937: 27904: 27896: 27858: 27850: 27800: 27790: 27757: 27747: 27714: 27606: 27580: 27572: 27538: 27504: 27436: 27428: 27356: 27313: 27130: 27018: 26991: 26882: 26845: 26785: 26756: 26724: 26716: 26653: 26645: 26441: 26390: 26354: 26313: 26305: 26236: 26195: 26154: 26094:449210 26092: 26041: 25975: 25938: 25901: 25868: 25825: 25790: 25753: 25710: 25676: 25592: 25584: 25576: 25523: 25492: 25452: 25444: 25436: 25386: 25328: 25282: 25257: 24721:subset 23680: 22664:. The 22333: 21479:is in 19685:Also, 19271:is of 18261:, and 17683: 17559:, and 16474:, and 14984:domain 13987:, the 13900:be an 13185: 12543: 12433: 12353: 12328: 12061:) and 11961: 11923: 11917: 11537: 10925: 10706: 10329: 9754:, and 9619: 9610: 7983: 6858: 6636: 6196: 5568: 5183:hold. 2541:) Let 1080:matrix 603:. The 528:> 1 453:> 1 291:or on 201: 138:domain 84:As in 31260:(PDF) 31212:(PDF) 31097:JSTOR 31069:JSTOR 31032:S2CID 30655:S2CID 30158:S2CID 30146:(2). 30084:S2CID 30072:(3). 30043:JSTOR 30008:JSTOR 29973:JSTOR 29831:S2CID 29729:S2CID 29665:88542 29661:JSTOR 29612:JSTOR 29577:JSTOR 29534:S2CID 29379:JSTOR 29287:JSTOR 29180:JSTOR 29145:JSTOR 29110:JSTOR 29034:JSTOR 28993:arXiv 28967:S2CID 28932:S2CID 28890:JSTOR 28848:S2CID 28813:S2CID 28795:arXiv 28672:S2CID 28623:arXiv 28621:(4). 28611:(PDF) 28590:S2CID 28547:S2CID 28529:arXiv 28502:S2CID 28474:arXiv 28447:S2CID 28405:S2CID 28099:(PDF) 28024:S2CID 28006:arXiv 27994:(PDF) 27902:S2CID 27848:JSTOR 27828:(PDF) 27798:S2CID 27755:S2CID 27712:S2CID 27694:arXiv 27604:JSTOR 27570:S2CID 27536:JSTOR 27502:S2CID 27426:S2CID 27354:S2CID 27311:S2CID 26909:(4). 26880:S2CID 26722:S2CID 26714:JSTOR 26525:(PDF) 26461:(PDF) 26439:S2CID 26421:arXiv 26388:S2CID 26234:S2CID 26193:S2CID 26090:S2CID 26070:arXiv 26039:JSTOR 25994:arXiv 25936:S2CID 25866:S2CID 25751:S2CID 25674:JSTOR 25490:S2CID 25434:JSTOR 25326:S2CID 25255:S2CID 24854:sheaf 24587:into 24421:over 24245:is a 23848:by a 23521:into 23138:is a 22518:chart 22116:is a 21940:into 21682:When 19926:germs 19275:over 18860:on a 18836:sheaf 17757:When 16057:is a 15867:is a 15100:with 14982:with 14426:, if 14400:When 14279:When 13391:with 12008:on a 12004:be a 11779:when 11292:is a 9746:then 9696:. 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18771:( 18765:) 18748:, 18741:f 18737:+ 18734:f 18731:( 18711:) 18708:I 18705:( 18699:) 18688:, 18681:f 18677:( 18674:, 18671:) 18665:, 18662:f 18659:( 18648:. 18636:) 18633:I 18630:( 18624:) 18607:, 18604:f 18601:a 18598:( 18578:) 18567:, 18564:a 18561:( 18541:) 18538:I 18535:( 18529:) 18523:, 18520:f 18517:( 18473:f 18453:) 18447:, 18444:f 18441:( 18421:) 18418:I 18415:( 18336:n 18331:C 18301:1 18295:5 18275:5 18269:4 18249:4 18243:1 18223:3 18217:2 18211:1 18197:D 18191:D 18187:. 18185:D 18177:D 18171:D 18165:D 18146:n 18141:C 18133:D 18063:n 18058:C 18046:Y 18042:X 18020:C 17997:) 17994:U 17991:( 17986:1 17958:Y 17952:y 17932:Y 17926:X 17923:: 17910:x 17906:f 17892:D 17886:U 17876:f 17872:x 17868:U 17854:D 17845:x 17815:n 17810:C 17794:D 17780:D 17774:) 17771:0 17768:( 17765:Q 17733:D 17727:) 17724:0 17721:( 17718:B 17698:) 17695:t 17689:0 17686:( 17680:D 17674:) 17671:t 17668:( 17665:Q 17645:} 17642:1 17639:= 17635:| 17631:u 17627:| 17623:; 17620:) 17617:t 17614:, 17611:u 17608:( 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16497:= 16491:{ 16488:= 16485:D 16462:} 16459:0 16450:{ 16447:= 16444:D 16422:2 16416:C 16392:R 16383:n 16378:C 16373:: 16360:D 16344:2 16338:C 16298:H 16278:) 16275:X 16272:( 16261:C 16252:) 16249:X 16246:( 16217:n 16213:X 16191:) 16188:X 16185:( 16174:C 16165:) 16162:X 16159:( 16138:n 16134:X 16120:) 16117:X 16114:( 16103:C 16094:) 16091:X 16088:( 16067:x 16063:X 16045:} 16042:x 16033:) 16030:z 16027:( 16021:; 16018:X 16012:z 16009:{ 15999:X 15965:X 15949:n 15943:C 15934:X 15916:u 15900:2 15894:C 15881:u 15855:f 15835:1 15820:u 15804:2 15798:C 15767:j 15757:z 15744:i 15740:z 15731:u 15726:2 15715:= 15710:j 15707:i 15699:, 15696:) 15691:j 15688:i 15680:( 15677:= 15672:u 15668:H 15644:u 15622:2 15616:C 15593:u 15573:0 15566:) 15555:z 15546:z 15538:u 15533:2 15522:( 15518:4 15515:= 15502:z 15488:) 15485:z 15482:( 15479:u 15476:= 15473:u 15449:C 15419:} 15410:{ 15403:R 15384:f 15361:X 15321:} 15312:{ 15305:R 15298:X 15292:f 15267:X 15239:. 15236:} 15233:D 15227:z 15224:b 15221:+ 15218:a 15215:; 15211:C 15204:z 15201:{ 15177:) 15174:z 15171:b 15168:+ 15165:a 15162:( 15159:f 15153:z 15127:n 15122:C 15114:b 15111:, 15108:a 15086:n 15081:C 15073:} 15069:C 15062:z 15059:; 15056:z 15053:b 15050:+ 15047:a 15044:{ 15008:n 15002:C 14993:D 14968:, 14965:} 14956:{ 14948:R 14940:D 14934:f 14903:. 14901:D 14887:) 14882:1 14878:z 14874:( 14870:R 14840:) 14835:1 14831:z 14827:( 14824:R 14817:| 14811:2 14807:z 14802:| 14781:D 14773:1 14769:z 14746:2 14741:C 14709:D 14705:R 14690:C 14679:D 14653:) 14648:n 14643:C 14635:D 14632:( 14612:D 14604:n 14599:C 14594:= 14589:c 14585:D 14564:) 14559:c 14555:D 14551:, 14548:K 14545:( 14530:D 14526:K 14512:) 14507:c 14503:D 14499:, 14494:D 14484:K 14477:( 14469:= 14466:) 14461:c 14457:D 14453:, 14450:K 14447:( 14432:D 14428:f 14414:1 14408:n 14385:G 14379:K 14373:G 14363:K 14347:G 14337:K 14313:G 14293:1 14290:= 14287:n 14270:G 14254:G 14244:K 14237:, 14234:K 14210:G 14197:G 14183:) 14180:G 14177:( 14172:O 14143:. 14139:} 14135:. 14132:) 14129:G 14126:( 14121:O 14113:f 14104:| 14100:) 14097:w 14094:( 14091:f 14087:| 14081:K 14075:w 14063:| 14059:) 14056:z 14053:( 14050:f 14046:| 14042:; 14039:G 14033:z 14029:{ 14020:G 14010:K 13993:K 13975:G 13969:K 13959:G 13945:) 13942:G 13939:( 13934:O 13908:n 13888:G 13866:n 13861:C 13853:G 13824:3 13818:n 13789:2 13785:D 13776:1 13772:D 13749:2 13745:D 13722:1 13718:D 13688:n 13684:D 13673:1 13670:= 13667:n 13659:= 13656:D 13628:2 13624:D 13615:1 13611:D 13581:D 13575:n 13570:1 13567:= 13556:= 13553:D 13531:n 13527:D 13523:, 13517:, 13512:1 13508:D 13465:C 13458:D 13438:1 13435:= 13432:n 13419:U 13405:g 13402:= 13399:f 13389:V 13385:g 13381:D 13377:f 13363:V 13357:D 13351:U 13331:D 13325:V 13303:n 13298:C 13290:V 13270:D 13264:U 13238:n 13233:C 13221:n 13217:D 13200:) 13197:2 13191:n 13188:( 13180:n 13175:C 13167:D 13157:D 13153:f 13149:D 13145:f 13141:f 13137:D 13123:D 13110:D 13094:n 13089:C 13081:D 13071:f 13064:D 13028:H 12998:n 12993:C 12969:2 12964:C 12952:H 12936:n 12931:C 12911:. 12897:2 12893:G 12889:= 12886:) 12881:1 12877:G 12873:( 12830:) 12827:0 12819:i 12815:r 12811:( 12806:) 12803:i 12800:( 12793:z 12787:i 12783:r 12774:i 12770:z 12747:n 12742:C 12732:n 12727:C 12722:: 12697:2 12693:G 12670:1 12666:G 12655:n 12625:) 12622:1 12616:, 12613:0 12607:p 12604:( 12600:} 12597:1 12588:p 12585:2 12579:| 12574:w 12570:| 12566:+ 12561:2 12556:| 12551:z 12547:| 12540:; 12535:2 12530:C 12522:) 12519:w 12516:, 12513:z 12510:( 12507:{ 12485:} 12482:1 12474:2 12469:| 12464:w 12460:| 12456:+ 12451:2 12446:| 12441:z 12437:| 12430:; 12425:2 12420:C 12412:) 12409:w 12406:, 12403:z 12400:( 12397:{ 12375:} 12372:1 12365:| 12361:w 12357:| 12350:, 12347:1 12340:| 12336:z 12332:| 12325:; 12320:2 12315:C 12307:) 12304:w 12301:, 12298:z 12295:( 12292:{ 12248:H 12225:2 12194:H 12167:H 12144:2 12113:H 12090:. 12088:G 12084:f 12079:K 12075:G 12067:G 12063:K 12057:n 12054:( 12040:n 12035:C 12023:G 12018:K 12014:G 12002:f 11982:) 11979:1 11967:0 11964:( 11958:} 11954:| 11948:2 11944:z 11939:| 11926:1 11907:| 11901:1 11897:z 11892:| 11888:; 11883:2 11872:) 11867:2 11863:z 11859:, 11854:1 11850:z 11846:( 11843:= 11840:z 11837:{ 11834:= 11825:H 11811:. 11799:1 11787:0 11767:} 11764:1 11757:| 11751:2 11747:z 11742:| 11738:, 11735:1 11728:| 11722:1 11718:z 11713:| 11709:; 11704:2 11699:C 11691:z 11688:{ 11685:= 11680:2 11649:n 11644:C 11609:0 11606:= 11603:a 11581:n 11577:z 11573:, 11567:, 11562:1 11558:z 11530:n 11526:k 11521:) 11515:n 11511:a 11502:n 11498:z 11494:( 11484:1 11480:k 11475:) 11469:1 11465:a 11456:1 11452:z 11448:( 11441:n 11437:k 11433:, 11427:, 11422:1 11418:k 11413:c 11402:0 11399:= 11394:n 11390:k 11386:, 11380:, 11375:1 11371:k 11343:n 11338:C 11311:n 11306:R 11277:) 11273:| 11267:n 11263:z 11258:| 11248:, 11242:, 11238:| 11232:1 11228:z 11223:| 11213:( 11210:= 11207:) 11204:z 11201:( 11192:z 11189:; 11160:} 11157:0 11149:n 11145:z 11141:, 11135:, 11130:1 11126:z 11122:; 11119:D 11113:) 11108:n 11104:z 11100:, 11094:, 11089:1 11085:z 11081:( 11078:= 11075:z 11072:{ 11069:= 11060:D 11036:) 11027:D 11023:( 11006:D 10979:a 10971:D 10954:. 10950:} 10946:n 10943:, 10937:, 10934:1 10931:= 10922:, 10918:| 10908:a 10899:0 10890:z 10885:| 10877:| 10867:a 10854:z 10849:| 10845:; 10842:) 10837:n 10833:z 10829:, 10823:, 10818:1 10814:z 10810:( 10807:= 10804:z 10800:{ 10776:D 10768:0 10764:z 10753:a 10749:D 10732:. 10728:} 10721:2 10709:0 10703:, 10695:i 10691:e 10686:) 10682:a 10674:0 10670:z 10665:( 10661:+ 10658:a 10655:= 10652:z 10649:; 10646:) 10641:n 10637:z 10633:, 10627:, 10622:1 10618:z 10614:( 10611:= 10608:z 10604:{ 10580:D 10572:0 10568:z 10543:} 10539:) 10530:a 10521:0 10512:z 10508:( 10493:i 10489:e 10484:{ 10476:} 10466:a 10457:0 10453:z 10448:{ 10437:D 10423:) 10420:n 10417:, 10411:, 10408:1 10405:= 10399:( 10375:D 10358:. 10354:} 10350:n 10347:, 10341:, 10338:1 10335:= 10326:, 10322:| 10312:a 10303:0 10294:z 10289:| 10285:= 10281:| 10271:a 10258:z 10253:| 10249:; 10246:) 10241:n 10237:z 10233:, 10227:, 10222:1 10218:z 10214:( 10211:= 10208:z 10204:{ 10180:D 10174:) 10169:0 10164:n 10160:z 10156:, 10150:, 10145:0 10140:1 10136:z 10132:( 10129:= 10124:0 10120:z 10097:n 10092:C 10084:) 10079:n 10075:a 10071:, 10065:, 10060:1 10056:a 10052:( 10049:= 10046:a 10026:1 10020:n 10012:( 9998:n 9993:C 9985:D 9963:W 9959:U 9955:V 9941:2 9935:n 9914:U 9891:) 9888:U 9885:( 9880:O 9872:f 9852:) 9849:V 9846:( 9841:O 9833:g 9823:U 9809:f 9799:V 9795:U 9781:U 9768:W 9764:g 9760:f 9756:g 9752:V 9748:f 9732:W 9727:| 9722:g 9719:= 9714:W 9709:| 9704:f 9684:V 9678:U 9654:W 9628:V 9622:U 9616:, 9613:V 9607:, 9604:U 9594:U 9580:) 9577:U 9574:( 9569:O 9547:) 9544:V 9541:( 9536:O 9528:g 9508:) 9505:U 9502:( 9497:O 9489:f 9467:n 9462:C 9408:1 9402:n 9384:V 9380:U 9376:V 9372:U 9338:U 9332:V 9329:: 9324:1 9296:V 9290:U 9287:: 9262:n 9257:C 9245:n 9241:V 9237:U 9197:0 9194:= 9189:1 9185:z 9162:2 9157:C 9133:1 9129:z 9125:= 9122:) 9117:2 9113:z 9109:, 9104:1 9100:z 9096:( 9093:g 9073:0 9070:= 9067:) 9062:2 9058:z 9054:, 9049:1 9045:z 9041:( 9038:f 9028:D 9024:D 9020:N 9004:n 8999:C 8991:D 8958:. 8955:) 8952:z 8949:, 8943:( 8937:) 8931:( 8928:f 8923:D 8912:= 8909:) 8906:z 8903:( 8900:f 8886:f 8868:. 8865:) 8862:z 8859:, 8853:( 8844:) 8838:( 8835:f 8820:D 8809:) 8806:z 8803:, 8797:( 8791:) 8785:( 8782:f 8777:D 8766:= 8763:) 8760:z 8757:( 8754:f 8730:n 8722:d 8714:n 8699:d 8685:j 8677:d 8663:1 8655:d 8647:1 8632:d 8628:) 8623:j 8614:z 8603:j 8588:( 8583:n 8577:j 8571:1 8558:n 8555:2 8550:| 8539:z 8535:| 8530:1 8520:n 8516:) 8512:i 8506:2 8503:( 8498:! 8495:) 8492:1 8486:n 8483:( 8477:= 8474:) 8471:z 8468:, 8462:( 8436:) 8433:1 8427:n 8424:, 8421:n 8418:( 8374:) 8371:z 8368:, 8362:( 8337:n 8332:C 8320:z 8296:D 8292:z 8254:D 8229:n 8224:C 8212:D 8208:f 8173:R 8157:R 8130:r 8114:r 8084:R 8076:| 8072:z 8068:| 8052:r 8024:) 8021:k 8018:= 8013:n 8005:+ 7999:+ 7994:1 7986:( 7976:k 7972:z 7968:1 7954:f 7950:d 7946:) 7942:0 7936:, 7931:1 7923:n 7913:n 7900:1 7892:1 7882:n 7867:! 7862:n 7851:! 7846:1 7836:! 7833:k 7826:, 7820:, 7817:0 7813:( 7806:) 7800:( 7780:r 7776:= 7772:| 7757:| 7745:i 7739:2 7735:1 7727:! 7724:k 7720:1 7708:1 7705:= 7702:k 7694:+ 7682:k 7678:z 7665:f 7661:d 7656:0 7653:= 7650:z 7645:] 7638:z 7628:1 7618:k 7614:z 7610:d 7604:k 7600:d 7593:[ 7585:) 7579:( 7559:R 7555:= 7551:| 7536:| 7522:n 7518:) 7514:i 7508:2 7505:( 7501:1 7493:! 7490:k 7486:1 7474:0 7471:= 7468:k 7460:= 7453:) 7450:z 7447:( 7416:} 7412:n 7409:, 7403:, 7400:1 7397:+ 7386:, 7377:R 7369:| 7365:z 7361:| 7348:r 7344:; 7339:n 7334:C 7326:) 7321:n 7317:z 7313:, 7307:, 7302:2 7298:z 7294:, 7289:1 7285:z 7281:( 7278:= 7275:z 7271:{ 7247:) 7244:z 7241:( 7205:} 7201:n 7198:, 7192:, 7189:1 7186:= 7175:, 7166:r 7158:| 7148:a 7135:z 7130:| 7126:; 7121:n 7116:C 7108:) 7103:n 7099:z 7095:, 7089:, 7084:2 7080:z 7076:, 7071:1 7067:z 7063:( 7060:= 7057:z 7053:{ 7031:} 7027:n 7024:, 7018:, 7015:1 7012:= 7001:, 6992:r 6984:| 6974:a 6961:z 6956:| 6952:; 6947:n 6942:C 6934:) 6929:n 6925:z 6921:, 6915:, 6910:2 6906:z 6902:, 6897:1 6893:z 6889:( 6886:= 6883:z 6879:{ 6851:n 6847:k 6842:) 6836:n 6832:a 6823:n 6819:z 6815:( 6805:1 6801:k 6796:) 6790:1 6786:a 6777:1 6773:z 6769:( 6762:n 6758:k 6754:, 6748:, 6743:1 6739:k 6734:c 6723:0 6720:= 6715:n 6711:k 6707:, 6701:, 6696:1 6692:k 6666:} 6663:) 6660:n 6657:, 6651:, 6648:1 6645:= 6639:( 6627:r 6623:{ 6588:n 6584:k 6576:n 6572:z 6556:1 6552:k 6544:1 6540:z 6527:v 6523:f 6515:n 6511:k 6507:+ 6501:+ 6496:1 6492:k 6473:1 6470:= 6467:v 6459:= 6449:n 6445:k 6437:n 6433:z 6417:1 6413:k 6405:1 6401:z 6390:f 6383:n 6379:k 6375:+ 6369:+ 6364:1 6360:k 6339:. 6337:f 6333:D 6317:v 6313:f 6302:D 6286:v 6282:f 6271:f 6267:D 6251:n 6247:f 6243:, 6237:, 6232:1 6228:f 6199:, 6189:n 6185:k 6180:) 6174:n 6170:a 6161:n 6157:z 6153:( 6143:1 6139:k 6134:) 6128:1 6124:a 6115:1 6111:z 6107:( 6100:n 6096:k 6092:, 6086:, 6081:1 6077:k 6072:c 6061:0 6058:= 6053:n 6049:k 6045:, 6039:, 6034:1 6030:k 6021:= 6018:) 6015:z 6012:( 6009:f 5996:D 5982:) 5979:z 5976:( 5973:f 5951:n 5946:C 5938:D 5932:) 5927:n 5923:a 5919:, 5913:, 5908:1 5904:a 5900:( 5897:= 5894:a 5881:f 5858:n 5850:d 5842:1 5834:d 5825:1 5822:+ 5817:n 5813:k 5808:) 5802:n 5798:a 5789:n 5781:( 5773:1 5770:+ 5765:1 5761:k 5756:) 5750:1 5746:a 5737:1 5729:( 5724:) 5719:n 5711:, 5705:, 5700:1 5692:( 5689:f 5679:n 5675:D 5656:1 5652:D 5635:n 5631:) 5627:i 5621:2 5618:( 5614:1 5609:= 5602:n 5598:k 5589:1 5585:k 5580:c 5571:, 5561:n 5557:k 5552:) 5546:n 5542:a 5533:n 5529:z 5525:( 5515:1 5511:k 5506:) 5500:1 5496:a 5487:1 5483:z 5479:( 5472:n 5468:k 5464:, 5458:, 5453:1 5449:k 5444:c 5433:0 5430:= 5425:n 5421:k 5417:, 5411:, 5406:1 5402:k 5393:= 5390:) 5387:z 5384:( 5381:f 5353:} 5350:n 5347:, 5341:, 5338:1 5335:= 5324:, 5315:r 5307:| 5297:a 5284:z 5279:| 5275:; 5270:n 5265:C 5257:) 5252:n 5248:z 5244:, 5238:, 5233:2 5229:z 5225:, 5220:1 5216:z 5212:( 5209:= 5206:z 5203:{ 5193:f 5157:n 5153:k 5145:n 5141:r 5128:1 5124:k 5116:1 5112:r 5104:! 5099:n 5095:k 5086:1 5082:k 5078:M 5068:| 5058:n 5054:k 5046:n 5042:z 5026:1 5022:k 5014:1 5010:z 4999:) 4994:n 4986:, 4980:, 4975:2 4967:, 4962:1 4954:( 4951:f 4944:n 4940:k 4936:+ 4930:+ 4925:1 4921:k 4909:| 4884:M 4876:| 4872:f 4868:| 4846:} 4842:n 4839:, 4833:, 4830:1 4827:= 4816:, 4807:r 4799:| 4789:z 4771:| 4767:; 4762:n 4757:C 4749:) 4744:n 4736:, 4730:, 4725:2 4717:, 4712:1 4704:( 4701:= 4694:{ 4683:f 4658:C 4645:f 4637:) 4635:2 4633:( 4616:. 4611:n 4603:d 4595:1 4587:d 4578:1 4575:+ 4570:n 4566:k 4561:) 4555:n 4551:z 4542:n 4534:( 4526:1 4523:+ 4518:1 4514:k 4509:) 4503:1 4499:z 4490:1 4482:( 4477:) 4472:n 4464:, 4458:, 4453:1 4445:( 4442:f 4432:n 4428:D 4409:1 4405:D 4388:n 4384:) 4380:i 4374:2 4371:( 4366:! 4361:n 4357:k 4348:1 4344:k 4337:= 4327:n 4323:k 4315:n 4311:z 4295:1 4291:k 4283:1 4279:z 4268:) 4263:n 4255:, 4249:, 4244:2 4236:, 4231:1 4223:( 4220:f 4213:n 4209:k 4205:+ 4199:+ 4194:1 4190:k 4165:1 4150:) 4148:1 4146:( 4127:n 4119:d 4111:1 4103:d 4096:) 4091:n 4087:z 4078:n 4070:( 4064:) 4059:1 4055:z 4046:1 4038:( 4033:) 4028:n 4020:, 4014:, 4009:1 4001:( 3998:f 3988:n 3984:D 3965:1 3961:D 3944:n 3940:) 3936:i 3930:2 3927:( 3923:1 3918:= 3915:) 3910:n 3906:z 3902:, 3896:, 3891:1 3887:z 3883:( 3880:f 3857:f 3843:D 3811:1 3803:d 3796:) 3791:n 3787:z 3778:n 3770:( 3764:) 3759:2 3755:z 3746:2 3738:( 3735:) 3730:1 3726:z 3717:1 3709:( 3704:) 3699:n 3691:, 3685:, 3680:2 3672:, 3667:1 3659:( 3656:f 3646:1 3642:D 3628:2 3620:d 3612:2 3608:D 3591:n 3583:d 3575:n 3571:D 3554:n 3550:) 3546:i 3540:2 3537:( 3533:1 3528:= 3516:1 3508:d 3501:) 3496:2 3492:z 3483:2 3475:( 3472:) 3467:1 3463:z 3454:1 3446:( 3441:) 3436:n 3432:z 3428:, 3422:, 3417:3 3413:z 3409:, 3404:2 3396:, 3391:1 3383:( 3380:f 3370:1 3366:D 3352:2 3344:d 3336:2 3332:D 3315:2 3311:) 3307:i 3301:2 3298:( 3294:1 3289:= 3277:1 3269:d 3260:1 3256:z 3247:1 3237:) 3232:n 3228:z 3224:, 3218:, 3213:2 3209:z 3205:, 3200:1 3192:( 3189:f 3179:1 3175:D 3160:i 3154:2 3150:1 3145:= 3138:) 3133:n 3129:z 3125:, 3119:, 3114:1 3110:z 3106:( 3103:f 3070:n 3065:1 3062:= 3055:} 3051:z 3048:{ 3027:} 3023:n 3020:, 3014:, 3011:1 3008:= 2991:r 2983:| 2973:z 2955:| 2951:; 2946:n 2941:C 2933:) 2928:n 2920:, 2914:, 2909:2 2901:, 2896:1 2888:( 2885:= 2878:{ 2874:= 2871:) 2868:r 2865:, 2862:z 2859:( 2826:n 2822:D 2807:2 2803:D 2794:1 2790:D 2725:D 2711:n 2707:D 2692:2 2688:D 2679:1 2675:D 2643:n 2639:D 2624:2 2620:D 2611:1 2607:D 2550:D 2529:n 2526:, 2520:, 2517:2 2514:, 2511:1 2508:= 2483:1 2477:C 2410:D 2406:f 2382:= 2379:f 2343:, 2340:0 2337:= 2324:i 2320:z 2309:f 2299:, 2296:} 2293:n 2290:, 2284:, 2281:1 2278:{ 2272:i 2238:i 2234:x 2225:v 2213:= 2205:i 2201:y 2192:u 2171:i 2167:y 2158:v 2149:= 2141:i 2137:x 2128:u 2118:, 2115:} 2112:n 2109:, 2103:, 2100:1 2097:{ 2091:i 2068:f 2048:v 2028:u 2007:C 1998:n 1993:C 1988:: 1985:f 1963:) 1960:p 1957:( 1951:x 1943:v 1931:= 1928:) 1925:p 1922:( 1916:y 1908:u 1892:) 1889:p 1886:( 1880:y 1872:v 1863:= 1860:) 1857:p 1854:( 1848:x 1840:u 1814:p 1790:v 1770:u 1749:C 1742:p 1721:C 1713:C 1709:: 1706:f 1684:f 1672:f 1668:f 1664:f 1648:) 1643:n 1639:z 1635:, 1629:, 1624:1 1621:+ 1618:i 1614:z 1610:, 1607:z 1604:, 1599:1 1593:i 1589:z 1585:, 1579:, 1574:1 1570:z 1566:( 1563:f 1557:z 1545:f 1538:D 1534:f 1518:) 1512:h 1506:( 1503:o 1500:+ 1497:) 1494:h 1491:( 1488:L 1485:+ 1482:) 1479:z 1476:( 1473:f 1470:= 1467:) 1464:h 1461:+ 1458:z 1455:( 1452:f 1429:C 1420:n 1415:C 1410:: 1407:L 1387:D 1381:z 1360:C 1337:n 1332:C 1324:D 1314:f 1285:n 1280:C 1256:n 1251:C 1206:. 1201:2 1196:| 1191:w 1187:| 1183:= 1178:2 1174:v 1170:+ 1165:2 1161:u 1131:, 1126:) 1120:u 1115:v 1108:v 1100:u 1094:( 1071:u 1067:w 1043:i 1030:I 1028:− 1024:J 1019:J 1004:n 1002:2 982:n 979:2 974:R 947:n 942:C 915:R 903:n 901:2 891:n 866:n 861:C 833:n 828:C 804:n 799:C 776:C 764:n 743:n 738:C 549:n 544:C 532:D 526:n 510:C 499:D 462:n 451:n 435:C 426:n 421:C 416:: 413:f 277:n 272:C 248:n 243:P 240:C 230:( 210:2 204:n 198:, 193:n 188:C 180:D 155:C 148:D 140:( 130:n 113:i 111:z 91:n 63:n 47:n 42:C

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Knowledge

Function of several complex variables

Index