20:
2955:. Many properties of a K3 surface are determined by its Picard lattice, as a symmetric bilinear form over the integers. This leads to a strong connection between the theory of K3 surfaces and the arithmetic of symmetric bilinear forms. As a first example of this connection: a complex analytic K3 surface is algebraic if and only if there is an element
259:
There are several equivalent ways to define K3 surfaces. The only compact complex surfaces with trivial canonical bundle are K3 surfaces and compact complex tori, and so one can add any condition excluding the latter to define K3 surfaces. For example, it is equivalent to define a complex analytic K3
5887:
gave K3 surfaces their name (see the quotation above) and made several influential conjectures about their classification. Kunihiko
Kodaira completed the basic theory around 1960, in particular making the first systematic study of complex analytic K3 surfaces which are not algebraic. He showed that
3374:.) It follows that having an elliptic fibration is a codimension-1 condition on a K3 surface. So there are 19-dimensional families of complex analytic K3 surfaces with an elliptic fibration, and 18-dimensional moduli spaces of projective K3 surfaces with an elliptic fibration.
5457:
of the roots form a set of hyperplanes which all go through the positive cone. Then the ample cone is a connected component of the complement of these hyperplanes in the positive cone. Any two such components are isomorphic via the orthogonal group of the lattice
5623:, there is one other possibility: the cone of curves may be spanned by one (â2)-curve and one curve with self-intersection 0.) So the cone of curves is either the standard round cone, or else it has "sharp corners" (because every (â2)-curve spans an
3998:
705:
5751:
heterotic string, the Spin(32)/Z2 heterotic string, and M-theory are related by compactification on a K3 surface. For example, the Type IIA string compactified on a K3 surface is equivalent to the heterotic string compactified on a 4-torus
3872:
835:
5888:
any two complex analytic K3 surfaces are deformation-equivalent and hence diffeomorphic, which was new even for algebraic K3 surfaces. An important later advance was the proof of the
Torelli theorem for complex algebraic K3 surfaces by
5635:
K3 surfaces are somewhat unusual among algebraic varieties in that their automorphism groups may be infinite, discrete, and highly nonabelian. By a version of the
Torelli theorem, the Picard lattice of a complex algebraic K3 surface
2186:. This would be optimal if true, since equality holds for a complex K3 surface, which has signature 3â19 = â16. The conjecture would imply that every simply connected smooth 4-manifold with even intersection form is
5239:
5447:
4255:
3670:
1101:
250:
It can be useful to think of complex algebraic K3 surfaces as part of the broader family of complex analytic K3 surfaces. Many other types of algebraic varieties do not have such non-algebraic deformations.
553:
2694:
2018:
1190:
1935:
5179:
of ample divisors (up to automorphisms of the Picard lattice). The ample cone is determined by the Picard lattice as follows. By the Hodge index theorem, the intersection form on the real vector space
1871:
1763:
of algebraic K3 surfaces to show that all such K3 surfaces have the same Hodge numbers. A more low-brow calculation can be done using the calculation of the Betti numbers along with the parts of the
2127:
1447:
4073:
3230:, with the possible types of singular fibers classified by Kodaira. There are always some singular fibers, since the sum of the topological Euler characteristics of the singular fibers is
3335:
2991:
2908:
3753:
has arbitrarily small deformations which are isomorphic to smooth quartics.) For the same reason, there is not a meaningful moduli space of compact complex tori of dimension at least 2.
2062:
191:
1240:
981:
4176:
4132:
3599:
3447:
3224:
2831:
1809:
342:
4304:
295:
of a complex analytic K3 surface are computed as follows. (A similar argument gives the same answer for the Betti numbers of an algebraic K3 surface over any field, defined using
33:
Dans la seconde partie de mon rapport, il s'agit des variétés kÀhlériennes dites K3, ainsi nommées en l'honneur de Kummer, KÀhler, Kodaira et de la belle montagne K2 au
Cachemire.
4840:
1578:
1514:
5006:
4975:
4944:
4909:
4878:
4754:
4649:
4569:
5087:
4396:
2232:
5844:
5791:
5157:
5116:
4532:
4495:
3539:
3408:
2549:
2517:
2466:
2341:
2305:
452:
399:
5591:
5559:
3152:
2787:
5513:
5277:
3743:
3714:
3263:
3126:
2953:
2384:
1620:
1358:
1023:
271:
There are also some variants of the definition. Over the complex numbers, some authors consider only the algebraic K3 surfaces. (An algebraic K3 surface is automatically
3910:
3024:
3469:; that is, it is not covered by a continuous family of rational curves. On the other hand, in contrast to negatively curved varieties such as surfaces of general type,
2437:
572:
5372:
5313:
5163:
is still irreducible of dimension 19 (containing the previous moduli space as an open subset). Formally, it works better to view this as a moduli space of K3 surfaces
4784:
4687:
3774:
3178:
3073:
1316:
1276:
874:
5870:
5681:
5621:
5533:
4719:
4599:
3368:
5535:
is empty, then the closed cone of curves is the closure of the positive cone. Otherwise, the closed cone of curves is the closed convex cone spanned by all elements
3901:
3290:
3047:
2870:
2738:
2718:
2631:
2158:
913:
720:
3453:. The moduli space of all smooth quartic surfaces (up to isomorphism) has dimension 19, while the subspace of quartic surfaces containing a line has dimension 18.
3512:
happens to be an elliptic K3 surface.) A stronger question that remains open is whether every complex K3 surface admits a nondegenerate holomorphic map from
2241:. On the other hand, there are smooth complex surfaces (some of them projective) that are homeomorphic but not diffeomorphic to a K3 surface, by Kodaira and
5743:
on these surfaces are not trivial, yet they are simple enough to analyze most of their properties in detail. The type IIA string, the type IIB string, the E
5008:. However, a concrete version of this idea is the fact that any two complex algebraic K3 surfaces are deformation-equivalent through algebraic K3 surfaces.
3749:. (For example, the space of smooth quartic surfaces is irreducible of dimension 19, and yet every complex analytic K3 surface in the 20-dimensional family
231:, and yet where a substantial understanding is possible. A complex K3 surface has real dimension 4, and it plays an important role in the study of smooth
3371:
7291:
6893:
4911:
corresponding to K3 surfaces of Picard number at least 2. Those K3 surfaces have polarizations of infinitely many different degrees, not just 2
5175:
A remarkable feature of algebraic K3 surfaces is that the Picard lattice determines many geometric properties of the surface, including the
3029:
Roughly speaking, the space of all complex analytic K3 surfaces has complex dimension 20, while the space of K3 surfaces with Picard number
5182:
223:(which are essentially unclassifiable). K3 surfaces can be considered the simplest algebraic varieties whose structure does not reduce to
5740:
2744:
contains no closed complex curves at all. By contrast, an algebraic surface always contains many continuous families of curves.) Over an
5380:
4181:
3604:
3226:. "Elliptic" means that all but finitely many fibers of this morphism are smooth curves of genus 1. The singular fibers are unions of
1034:
6780:"Géométrie des espaces de modules de courbes et de surfaces K3 (d'aprÚs Gritsenko-Hulek-Sankaran, Farkas-Popa, Mukai, Verra, et al.)"
2469:
464:
3082:
The precise description of which lattices can occur as Picard lattices of K3 surfaces is complicated. One clear statement, due to
2640:
1964:
1110:
5597:
contains no (â2)-curves; in the second case, the closed cone of curves is the closed convex cone spanned by all (â2)-curves. (If
1876:
1814:
215:) of dimension two. As such, they are at the center of the classification of algebraic surfaces, between the positively curved
6344:
6311:
3768:
holds: a K3 surface is determined by its Hodge structure. The period domain is defined as the 20-dimensional complex manifold
3075:(excluding the supersingular case). In particular, algebraic K3 surfaces occur in 19-dimensional families. More details about
6827:
6794:
6728:
6284:
2067:
711:
3295:
Whether a K3 surface is elliptic can be read from its Picard lattice. Namely, in characteristic not 2 or 3, a K3 surface
1666:
1369:
108:
6886:
6568:
6267:, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics, vol. 4,
3087:
4018:
3680:
complex manifold of dimension 20. The set of isomorphism classes of complex analytic K3 surfaces is the quotient of
7258:
3188:
An important subclass of K3 surfaces, easier to analyze than the general case, consists of the K3 surfaces with an
2607:
559:
3302:
2958:
2875:
7488:
7223:
5645:
240:
2034:
8086:
7850:
5846:
with trivial canonical bundle and irregularity zero. In 1909, Enriques showed that such surfaces exist for all
5316:
124:
7430:
6936:
6879:
6860:
6720:
6708:
4880:
overlap in an intricate way. Indeed, there is a countably infinite set of codimension-1 subvarieties of each
1195:
6779:
5466:
across each root hyperplane. In this sense, the Picard lattice determines the ample cone up to isomorphism.
918:
7875:
7200:
6931:
4137:
4093:
3560:
3417:
3194:
2792:
2487:
1770:
302:
4260:
8076:
6749:
6703:
4789:
4610:
2391:
2238:
1519:
1455:
7455:
4980:
4949:
4918:
4883:
4852:
4728:
4623:
4543:
7376:
7081:
7028:
5034:
4343:
2745:
2197:
27:(of real dimension 2) in a certain complex K3 surface (of complex dimension 2, hence real dimension 4).
5820:
5767:
5133:
5092:
4508:
4471:
3515:
3461:
In contrast to positively curved varieties such as del Pezzo surfaces, a complex algebraic K3 surface
3384:
3154:
is the Picard lattice of some complex projective K3 surface. The space of such surfaces has dimension
2525:
2493:
2442:
2317:
2281:
8081:
7670:
4426:
3504:
is always covered by a continuous family of images of elliptic curves. (These curves are singular in
2555:
402:
89:
7715:
2767:) together with its intersection form, a symmetric bilinear form with values in the integers. (Over
2394:
of this singular surface may also be called a Kummer surface; that resolution is a K3 surface. When
2237:
Every complex surface that is diffeomorphic to a K3 surface is a K3 surface, by Robert
Friedman and
408:
355:
7895:
7815:
7630:
7564:
6926:
6497:
5564:
5463:
3993:{\displaystyle H^{0}(X,\Omega ^{2})\subset H^{2}(X,\mathbb {C} )\cong \Lambda \otimes \mathbb {C} }
2753:
1938:
197:
7775:
5538:
3131:
2770:
2182:
with even intersection form has second Betti number at least 11/8 times the absolute value of the
700:{\displaystyle \chi (X,{\mathcal {O}}_{X}):=\sum _{i}(-1)^{i}h^{i}(X,{\mathcal {O}}_{X})=1-0+1=2.}
37:
In the second part of my report, we deal with the KĂ€hler varieties known as K3, named in honor of
8035:
7845:
7559:
7397:
7371:
7243:
7112:
7001:
6943:
6330:
5492:
5244:
3867:{\displaystyle D=\{u\in P(\Lambda \otimes \mathbb {C} ):u^{2}=0,\,u\cdot {\overline {u}}>0\}.}
3719:
3690:
3233:
3093:
2920:
2360:
2021:
1583:
1321:
986:
280:
208:
7402:
6698:
2996:
212:
8091:
7800:
7541:
7347:
7238:
7210:
7033:
6654:
5889:
3746:
2405:
5341:
5282:
4977:. This is imprecise, since there is not a well-behaved space containing all the moduli spaces
4763:
4666:
3541:(where "nondegenerate" means that the derivative of the map is an isomorphism at some point).
3157:
3052:
2696:. It is an important feature of K3 surfaces that many different Picard numbers can occur. For
1285:
1245:
843:
830:{\displaystyle \chi (X,{\mathcal {O}}_{X})={\frac {1}{12}}\left(c_{1}(X)^{2}+c_{2}(X)\right),}
7655:
7595:
7536:
7503:
7498:
7296:
7286:
7253:
7117:
6994:
6989:
6984:
6969:
6959:
5849:
5666:
5600:
5518:
5454:
4692:
4578:
4000:. This is surjective, and a local isomorphism, but not an isomorphism (in particular because
3340:
2266:
branched along a smooth sextic (degree 6) curve is a K3 surface of genus 2 (that is, degree 2
236:
7950:
6555:
3880:
1650:. (Analogously, but much easier: every algebraic K3 surface over a field is projective.) By
8040:
7825:
7038:
7023:
6979:
6856:
6837:
6804:
6738:
6690:
6670:
6647:
6620:
6578:
6536:
6516:
6430:
6384:
6352:
6319:
6294:
6249:
6028:
4418:
4178:, that is, an isomorphism of abelian groups that preserves the intersection form and sends
3268:
3032:
2848:
2723:
2703:
2616:
2183:
2136:
1361:
1029:
891:
276:
6768:
6484:
6457:
5892:
and Igor
Shafarevich (1971), extended to complex analytic K3 surfaces by Daniel Burns and
3716:, but this quotient is not a geometrically meaningful moduli space, because the action of
2171:
8:
7955:
7840:
7493:
7392:
7018:
5374:. Then the ample cone is equal to the positive cone. Thus it is the standard round cone.
4538:
3486:
2914:
2838:
2834:
2308:
93:
6865:
6674:
6563:, Cambridge Studies in Advanced Mathematics, vol. 158, Cambridge University Press,
6520:
6305:
1279:
8000:
7920:
7820:
7780:
7710:
7660:
7625:
7460:
7337:
7233:
6974:
6624:
6598:
6540:
6506:
6404:
6388:
6235:
5724:
3189:
3083:
2029:
272:
7387:
7965:
7870:
7705:
7615:
7585:
7381:
7276:
7228:
7122:
6823:
6790:
6724:
6564:
6551:
6464:
6437:
6280:
6227:
5919:
5806:
5159:
with self-intersection â2.) The moduli space of quasi-polarized K3 surfaces of genus
4330:
3473:
contains a large discrete set of rational curves (possibly singular). In particular,
2592:
2256:
1655:
296:
265:
228:
216:
112:
104:
6682:
6628:
6544:
6392:
7975:
7910:
7880:
7760:
7700:
7665:
7610:
7600:
7513:
7465:
7423:
7328:
7321:
7314:
7307:
7300:
7218:
7008:
6916:
6764:
6745:
6678:
6658:
6608:
6524:
6480:
6453:
6416:
6400:
6376:
6372:
6326:
6301:
6272:
5915:
5905:
5893:
5873:
4652:
3685:
3493:
3466:
2399:
2263:
2242:
1945:
1636:
261:
85:
81:
46:
7580:
6642:, Adv. Stud. Pure Math., vol. 45, Tokyo: Math. Soc. Japan, pp. 315â326,
1643:
8055:
8010:
7960:
7945:
7935:
7830:
7795:
7620:
7190:
6964:
6833:
6819:
6800:
6734:
6686:
6643:
6616:
6586:
6574:
6532:
6426:
6380:
6348:
6334:
6315:
6290:
6268:
6245:
5910:
5802:
5688:
5024:
5020:
4606:
3765:
3761:
3677:
3474:
2351:
2167:
1764:
1756:
224:
116:
100:
7900:
6495:; Sankaran, G. K. (2007), "The Kodaira dimension of the moduli of K3 surfaces",
6360:
42:
8030:
8025:
7985:
7925:
7755:
7745:
7740:
7735:
7650:
7645:
7640:
7605:
7590:
7518:
7195:
7058:
5736:
5478:
4414:
3757:
3227:
2347:
1752:
1651:
1104:
885:
24:
7915:
7835:
6528:
6276:
8070:
8020:
8005:
7980:
7970:
7940:
7885:
7860:
7790:
7785:
7750:
7725:
7685:
7475:
7127:
7043:
6921:
6902:
6775:
6635:
6338:
5798:
5469:
A related statement, due to SĂĄndor KovĂĄcs, is that knowing one ample divisor
4656:
3478:
3265:. A general elliptic K3 surface has exactly 24 singular fibers, each of type
2191:
1632:
455:
264:
compact complex manifold of dimension 2 with a nowhere-vanishing holomorphic
244:
111:
of surfaces, K3 surfaces form one of the four classes of minimal surfaces of
97:
7805:
6811:
268:. (The latter condition says exactly that the canonical bundle is trivial.)
8050:
7890:
7770:
7720:
7690:
7675:
7483:
7450:
7342:
7268:
7248:
7185:
7048:
6088:
Kamenova et al. (2014), Corollary 2.2; Huybrechts (2016), Corollary 13.2.2.
5794:
4757:
4602:
3370:. (In characteristic 2 or 3, the latter condition may also correspond to a
3076:
2568:
2187:
1760:
1647:
292:
220:
204:
38:
6612:
2789:, the intersection form means the restriction of the intersection form on
8045:
8015:
7995:
7855:
7810:
7765:
7730:
7680:
7445:
7414:
7175:
7132:
6492:
6260:
5176:
4722:
4660:
2917:
implies that the Picard lattice of an algebraic K3 surface has signature
1958:
881:
73:
6421:
5876:
showed that the moduli space of such surfaces has dimension 19 for each
2554:
There are several databases of K3 surfaces with du Val singularities in
7990:
7930:
7865:
7528:
7508:
7366:
7180:
6468:
6441:
6256:
6240:
5118:, but now it may contract finitely many (â2)-curves, so that the image
2833:. Over a general field, the intersection form can be defined using the
2275:
2161:
1659:
232:
7905:
7695:
7635:
7281:
7155:
7076:
7071:
7066:
6511:
5234:{\displaystyle N^{1}(X):=\operatorname {Pic} (X)\otimes \mathbb {R} }
4340:
is primitive (that is, not 2 or more times another line bundle) and
3299:
has an elliptic fibration if and only if there is a nonzero element
7551:
7440:
7435:
7165:
7160:
7097:
7013:
6340:
Géométrie des surfaces K3: modules et périodes, Séminaire
Palaiseau
4915:â2. So one can say that infinitely many of the other moduli spaces
2720:
can be any integer between 1 and 20. In the complex analytic case,
2591:. The two definitions agree for a complex algebraic K3 surface, by
2175:
219:(which are easy to classify) and the negatively curved surfaces of
6603:
23:
A smooth quartic surface in 3-space. The figure shows part of the
7170:
7150:
7107:
7102:
6850:
6020:
5442:{\displaystyle \Delta =\{u\in \operatorname {Pic} (X):u^{2}=-2\}}
4250:{\displaystyle H^{0}(X,\Omega ^{2})\subset H^{2}(X,\mathbb {C} )}
3500:
is identically zero. The proof uses that an algebraic K3 surface
2837:
of curves on a surface, by identifying the Picard group with the
1811:
for an arbitrary K3 surface. In this case, Hodge symmetry forces
54:
6871:
3665:{\displaystyle \Lambda =E_{8}(-1)^{\oplus 2}\oplus U^{\oplus 3}}
1096:{\displaystyle 0\to \mathbb {Z} _{X}\to O_{X}\to O_{X}^{*}\to 0}
6661:(1971), "Torelli's theorem for algebraic surfaces of type K3",
4078:
is bijective. It follows that two complex analytic K3 surfaces
2386:. This results in 16 singularities, at the 2-torsion points of
67:, p. 546), describing the reason for the name "K3 surface"
7142:
548:{\displaystyle h^{2}(X,{\mathcal {O}}_{X})=h^{0}(X,K_{X})=1.}
19:
6325:
3492:
Another contrast to negatively curved varieties is that the
2689:{\displaystyle H^{2}(X,\mathbb {Z} )\cong \mathbb {Z} ^{22}}
2579:
means the abelian group of complex analytic line bundles on
2245:. These "homotopy K3 surfaces" all have Kodaira dimension 1.
2013:{\displaystyle H^{2}(X,\mathbb {Z} )\cong \mathbb {Z} ^{22}}
1185:{\displaystyle 0\to H^{1}(X,\mathbb {Z} )\to H^{1}(X,O_{X})}
6750:"Le superficie algebriche con curva canonica d'ordine zero"
6589:(2014), "Kobayashi pseudometric on hyperkÀhler manifolds",
5739:
and provide an important tool for the understanding of it.
2596:
2270:â2 = 2). (This terminology means that the inverse image in
1930:{\displaystyle H^{1}(X,\Omega _{X})\cong \mathbb {C} ^{20}}
6255:
4329:
is defined to be a projective K3 surface together with an
4309:
1866:{\displaystyle H^{0}(X;\Omega _{X}^{2})\cong \mathbb {C} }
1658:, it follows that every complex analytic K3 surface has a
6789:, SĂ©minaire Bourbaki. 2006/2007. Exp 981 (317): 467â490,
6653:
5918:, a mysterious relationship between K3 surfaces and the
5170:
4497:. In most cases, this morphism is an embedding, so that
1755:
of a specific K3 surface, and then using a variation of
2402:
of a curve of genus 2, Kummer showed that the quotient
2122:{\displaystyle E_{8}(-1)^{\oplus 2}\oplus U^{\oplus 3}}
50:
6490:
6052:
Huybrechts (2016), Corollary 14.3.1 and Remark 14.3.7.
5715:. A related statement, due to Hans Sterk, is that Aut(
5852:
5823:
5770:
5669:
5603:
5567:
5541:
5521:
5495:
5383:
5344:
5285:
5247:
5185:
5136:
5095:
5037:
4983:
4952:
4921:
4886:
4855:
4792:
4766:
4731:
4695:
4669:
4626:
4581:
4546:
4511:
4474:
4346:
4263:
4184:
4140:
4096:
4021:
3913:
3883:
3777:
3722:
3693:
3607:
3563:
3518:
3489:
to a positive linear combination of rational curves.
3420:
3387:
3343:
3305:
3271:
3236:
3197:
3160:
3134:
3096:
3055:
3035:
2999:
2961:
2923:
2878:
2851:
2795:
2773:
2726:
2706:
2643:
2619:
2528:
2496:
2479:
with du Val singularities, the minimal resolution of
2445:
2408:
2363:
2320:
2284:
2200:
2139:
2070:
2037:
1967:
1879:
1817:
1773:
1586:
1522:
1458:
1372:
1324:
1288:
1248:
1198:
1113:
1037:
989:
921:
894:
846:
723:
575:
467:
411:
358:
305:
127:
16:
Type of smooth complex surface of kodaira dimension 0
6442:"Richerche di geometria sulle superficie algebriche"
6263:; Peters, Chris A.M.; Van de Ven, Antonius (2004) ,
6638:(2006), "Polarized K3 surfaces of genus thirteen",
6584:
6409:
6405:"On the Torelli problem for kÀhlerian K-3 surfaces"
4012:for K3 surfaces says that the quotient map of sets
1944:> 0, this was first shown by Alexey Rudakov and
1669:of any K3 surface are listed in the Hodge diamond:
1442:{\displaystyle \chi (X)=\sum _{i}(-1)^{i}b_{i}(X).}
286:
6814:(1958), "Final report on contract AF 18(603)-57",
6169:Huybrechts (2016), section 5.1.4 and Remark 6.4.5.
6115:Huybrechts (2016), section 6.3.1 and Remark 6.3.6.
5864:
5838:
5805:and other 19th-century geometers. More generally,
5785:
5675:
5615:
5585:
5553:
5527:
5507:
5441:
5366:
5307:
5271:
5233:
5151:
5110:
5081:
5000:
4969:
4938:
4903:
4872:
4834:
4778:
4748:
4713:
4681:
4643:
4593:
4563:
4526:
4489:
4390:
4298:
4249:
4170:
4126:
4067:
3992:
3895:
3866:
3737:
3708:
3664:
3593:
3533:
3441:
3402:
3362:
3329:
3284:
3257:
3218:
3172:
3146:
3120:
3067:
3041:
3018:
2985:
2947:
2902:
2864:
2825:
2781:
2732:
2712:
2688:
2625:
2543:
2511:
2460:
2431:
2378:
2335:
2299:
2226:
2152:
2121:
2056:
2012:
1929:
1865:
1803:
1614:
1572:
1508:
1441:
1352:
1310:
1270:
1234:
1184:
1095:
1017:
975:
907:
868:
829:
699:
547:
446:
393:
336:
185:
5323:the component that contains any ample divisor on
3456:
2752:> 0, there is a special class of K3 surfaces,
8068:
5663:)) generated by reflections in the set of roots
4398:. This is also called a polarized K3 surface of
2841:.) The Picard lattice of a K3 surface is always
6232:Fields, strings and duality (Boulder, CO, 1996)
5089:. Such a line bundle still gives a morphism to
5019:means a projective K3 surface with a primitive
2587:) means the group of algebraic line bundles on
2551:is a K3 surface of genus 5 (that is, degree 8).
2519:is a K3 surface of genus 4 (that is, degree 6).
2343:is a K3 surface of genus 3 (that is, degree 4).
4068:{\displaystyle N/O(\Lambda )\to D/O(\Lambda )}
6887:
6398:
5730:
5436:
5390:
3858:
3784:
3330:{\displaystyle u\in \operatorname {Pic} (X)}
2986:{\displaystyle u\in \operatorname {Pic} (X)}
2903:{\displaystyle u\in \operatorname {Pic} (X)}
2763:of a K3 surface means the abelian group Pic(
4849:The different 19-dimensional moduli spaces
2610:free abelian group; its rank is called the
283:of dimension 2), rather than being smooth.
107:that satisfies the same conditions. In the
6894:
6880:
6591:Journal of the London Mathematical Society
6550:
6230:(1996), "K3 surfaces and string duality",
5809:observed in 1893 that for various numbers
5735:K3 surfaces appear almost ubiquitously in
4571:of polarized complex K3 surfaces of genus
3090:, is that every even lattice of signature
2057:{\displaystyle \operatorname {II} _{3,19}}
2024:with values in the integers, known as the
6602:
6510:
6420:
6300:
6239:
6226:
5753:
5703:is commensurable with the quotient group
5279:. It follows that the set of elements of
5227:
5130:on a surface means a curve isomorphic to
4240:
4161:
4117:
4086:are isomorphic if and only if there is a
3986:
3969:
3835:
3806:
3584:
3521:
2816:
2775:
2676:
2664:
2000:
1988:
1917:
1859:
1794:
1751:One way to show this is to calculate the
1631:Any two complex analytic K3 surfaces are
1219:
1140:
1046:
186:{\displaystyle x^{4}+y^{4}+z^{4}+w^{4}=0}
92:zero. An (algebraic) K3 surface over any
6463:
6436:
5655:be the subgroup of the orthogonal group
5315:with positive self-intersection has two
2475:More generally: for any quartic surface
2133:is the hyperbolic lattice of rank 2 and
1642:Every complex analytic K3 surface has a
275:.) Or one may allow K3 surfaces to have
18:
6714:
6696:
4310:Moduli spaces of projective K3 surfaces
3676:of marked complex K3 surfaces is a non-
3183:
2314:A smooth quartic (degree 4) surface in
1235:{\displaystyle H^{1}(X,\mathbb {Z} )=0}
299:.) By definition, the canonical bundle
8069:
6774:
6744:
6307:Bourbaki seminar, Vol. 1982/83 Exp 609
6234:, World Scientific, pp. 421â540,
6070:Huybrechts (2016), Proposition 11.1.3.
4843:
4501:is isomorphic to a surface of degree 2
3557:to be an isomorphism of lattices from
3377:Example: Every smooth quartic surface
2522:The intersection of three quadrics in
976:{\displaystyle c_{1}(K_{X})=-c_{1}(X)}
558:As a result, the arithmetic genus (or
203:Together with two-dimensional compact
6875:
6640:Moduli spaces and arithmetic geometry
6634:
6361:"A database of polarized K3 surfaces"
6358:
5992:Huybrechts (2016), Proposition 3.3.5.
5640:determines the automorphism group of
5630:
5627:extremal ray of the cone of curves).
5171:The ample cone and the cone of curves
4842:. A survey of this area was given by
4421:implies that there is a smooth curve
4171:{\displaystyle H^{2}(Y,\mathbb {Z} )}
4127:{\displaystyle H^{2}(X,\mathbb {Z} )}
3594:{\displaystyle H^{2}(X,\mathbb {Z} )}
3442:{\displaystyle X\to \mathbf {P} ^{1}}
3219:{\displaystyle X\to \mathbf {P} ^{1}}
2826:{\displaystyle H^{2}(X,\mathbb {Z} )}
2562:
2350:is the quotient of a two-dimensional
1804:{\displaystyle H^{2}(X;\mathbb {Z} )}
337:{\displaystyle K_{X}=\Omega _{X}^{2}}
115:zero. A simple example is the Fermat
7292:Bogomol'nyiâPrasadâSommerfield bound
6868:, lectures by David Morrison (1988).
6810:
6343:, Astérisque, vol. 126, Paris:
6310:, Astérisque, vol. 105, Paris:
6187:Huybrechts (2016), Corollary 8.3.12.
6178:Huybrechts (2016), Corollary 8.2.11.
6142:Huybrechts (2016), Definition 2.4.1.
6079:Huybrechts (2016), Corollary 13.1.5.
6010:Huybrechts (2016), Remark 1.3.6(ii).
5884:
4299:{\displaystyle H^{0}(Y,\Omega ^{2})}
64:
6151:Huybrechts (2016), Corollary 6.4.4.
4835:{\displaystyle g=47,51,55,58,59,61}
2583:. For an algebraic K3 surface, Pic(
2575:) of a complex analytic K3 surface
2194:of copies of the K3 surface and of
1573:{\displaystyle b_{1}(X)=b_{3}(X)=0}
1509:{\displaystyle b_{0}(X)=b_{4}(X)=1}
235:. K3 surfaces have been applied to
13:
6818:, vol. II, Berlin, New York:
6816:Scientific works. Collected papers
6097:Huybrechts (2016), section 13.0.3.
6061:Huybrechts (2016), Remark 11.1.12.
5974:Barth et al. (2004), section IV.3.
5699:)), and the automorphism group of
5670:
5548:
5522:
5384:
5001:{\displaystyle {\mathcal {F}}_{g}}
4987:
4970:{\displaystyle {\mathcal {F}}_{g}}
4956:
4939:{\displaystyle {\mathcal {F}}_{h}}
4925:
4904:{\displaystyle {\mathcal {F}}_{g}}
4890:
4873:{\displaystyle {\mathcal {F}}_{g}}
4859:
4749:{\displaystyle {\mathcal {F}}_{g}}
4735:
4644:{\displaystyle {\mathcal {F}}_{g}}
4630:
4564:{\displaystyle {\mathcal {F}}_{g}}
4550:
4284:
4205:
4059:
4036:
3979:
3934:
3799:
3729:
3700:
3608:
1953:For a complex analytic K3 surface
1900:
1838:
915:is trivial, its first Chern class
739:
659:
591:
490:
320:
14:
8103:
6901:
6844:
6663:Mathematics of the USSR-Izvestiya
6196:Huybrechts (2016), Theorem 8.4.2.
6160:Huybrechts (2016), section 7.1.1.
6133:Huybrechts (2016), Theorem 7.5.3.
6124:Huybrechts (2016), section 7.1.3.
6106:Huybrechts (2016), section 6.3.3.
6043:Barth et al. (2004), Theorem 6.1.
5983:Huybrechts (2016), Theorem 9.5.1.
5965:Huybrechts (2016), Theorem 7.1.1.
5082:{\displaystyle c_{1}(L)^{2}=2g-2}
4725:and Gregory Sankaran showed that
4721:. In contrast, Valery Gritsenko,
4659:showed that this moduli space is
4655:complex variety of dimension 19.
4391:{\displaystyle c_{1}(L)^{2}=2g-2}
3553:of a complex analytic K3 surface
3544:
3496:on a complex analytic K3 surface
2740:may also be zero. (In that case,
2602:The Picard group of a K3 surface
2227:{\displaystyle S^{2}\times S^{2}}
2028:. This is isomorphic to the even
344:is trivial, and the irregularity
6585:Kamenova, Ljudmila; Lu, Steven;
5839:{\displaystyle \mathbf {P} ^{g}}
5826:
5813:, there are surfaces of degree 2
5786:{\displaystyle \mathbf {P} ^{3}}
5773:
5152:{\displaystyle \mathbf {P} ^{1}}
5139:
5111:{\displaystyle \mathbf {P} ^{g}}
5098:
4527:{\displaystyle \mathbf {P} ^{g}}
4514:
4490:{\displaystyle \mathbf {P} ^{g}}
4477:
4452:The vector space of sections of
3534:{\displaystyle \mathbb {C} ^{2}}
3449:, given by projecting away from
3429:
3403:{\displaystyle \mathbf {P} ^{3}}
3390:
3206:
3079:of K3 surfaces are given below.
2700:a complex algebraic K3 surface,
2544:{\displaystyle \mathbf {P} ^{5}}
2531:
2512:{\displaystyle \mathbf {P} ^{4}}
2499:
2461:{\displaystyle \mathbf {P} ^{3}}
2448:
2336:{\displaystyle \mathbf {P} ^{3}}
2323:
2300:{\displaystyle \mathbf {P} ^{2}}
2287:
560:holomorphic Euler characteristic
287:Calculation of the Betti numbers
7489:Eleven-dimensional supergravity
6683:10.1070/IM1971v005n03ABEH001075
6473:Rendiconti Accademia di Bologna
6214:Enriques (1909); Severi (1909).
6208:
6205:Enriques (1893), section III.6.
6199:
6190:
6181:
6172:
6163:
6154:
6145:
6136:
6127:
6118:
6109:
6100:
6091:
6082:
6073:
6064:
6055:
6046:
6037:
6013:
5956:Huybrechts (2016), section 2.4.
5947:Huybrechts (2016), section 2.3.
5938:Huybrechts (2016), Remark 1.1.2
109:EnriquesâKodaira classification
6377:10.1080/10586458.2007.10128983
6345:Société Mathématique de France
6312:Société Mathématique de France
6004:
5995:
5986:
5977:
5968:
5959:
5950:
5941:
5932:
5411:
5405:
5302:
5296:
5266:
5248:
5220:
5214:
5202:
5196:
5055:
5048:
4433:|. All such curves have genus
4364:
4357:
4293:
4274:
4244:
4230:
4214:
4195:
4165:
4151:
4121:
4107:
4062:
4056:
4042:
4039:
4033:
3973:
3959:
3943:
3924:
3887:
3810:
3796:
3732:
3726:
3703:
3697:
3634:
3624:
3588:
3574:
3457:Rational curves on K3 surfaces
3424:
3324:
3318:
3246:
3240:
3201:
3115:
3097:
2980:
2974:
2942:
2924:
2897:
2891:
2820:
2806:
2668:
2654:
2426:
2417:
2367:
2091:
2081:
1992:
1978:
1909:
1890:
1852:
1828:
1798:
1784:
1603:
1597:
1561:
1555:
1539:
1533:
1497:
1491:
1475:
1469:
1433:
1427:
1408:
1398:
1382:
1376:
1341:
1335:
1305:
1299:
1265:
1259:
1223:
1209:
1179:
1160:
1147:
1144:
1130:
1117:
1087:
1069:
1056:
1041:
1006:
1000:
970:
964:
945:
932:
863:
857:
816:
810:
788:
781:
750:
727:
670:
647:
628:
618:
602:
579:
536:
517:
501:
478:
447:{\displaystyle H^{1}(X,O_{X})}
441:
422:
394:{\displaystyle h^{1}(X,O_{X})}
388:
369:
84:of dimension 2 with Đ° trivial
49:and of the beautiful mountain
1:
6937:Second superstring revolution
6861:Magma computer algebra system
6851:Graded Ring Database homepage
6822:, pp. 390â395, 545â547,
6721:American Mathematical Society
6717:The wild world of 4-manifolds
6469:"Le superficie di genere uno"
6220:
5586:{\displaystyle A\cdot u>0}
3764:. When stated carefully, the
2468:as a quartic surface with 16
1625:
254:
7431:Generalized complex manifold
6932:First superstring revolution
6853:for a catalog of K3 surfaces
6001:Scorpan (2005), section 5.3.
5554:{\displaystyle u\in \Delta }
5330:Case 1: There is no element
3847:
3147:{\displaystyle \rho \leq 11}
2782:{\displaystyle \mathbb {C} }
1957:, the intersection form (or
1360:is equal to the topological
7:
6866:The geometry of K3 surfaces
6715:Scorpan, Alexandru (2005),
6704:Encyclopedia of Mathematics
6446:Memorie Accademia di Torino
5899:
5723:with a rational polyhedral
5508:{\displaystyle \rho \geq 3}
5462:), since that contains the
5453:of the Picard lattice. The
5272:{\displaystyle (1,\rho -1)}
5167:with du Val singularities.
3738:{\displaystyle O(\Lambda )}
3709:{\displaystyle O(\Lambda )}
3481:showed that every curve on
3258:{\displaystyle \chi (X)=24}
3121:{\displaystyle (1,\rho -1)}
2948:{\displaystyle (1,\rho -1)}
2845:, meaning that the integer
2633:. In the complex case, Pic(
2483:is an algebraic K3 surface.
2379:{\displaystyle a\mapsto -a}
2249:
2174:predicts that every smooth
1615:{\displaystyle b_{2}(X)=22}
1353:{\displaystyle c_{2}(X)=24}
1018:{\displaystyle c_{2}(X)=24}
714:(Noether's formula) says:
10:
8108:
7029:Non-critical string theory
5759:
5731:Relation to string duality
5719:) acts on the nef cone of
4417:. In characteristic zero,
3903:sends a marked K3 surface
3760:sends a K3 surface to its
3414:has an elliptic fibration
3019:{\displaystyle u^{2}>0}
2746:algebraically closed field
2556:weighted projective spaces
1635:as smooth 4-manifolds, by
198:complex projective 3-space
7573:
7550:
7527:
7474:
7359:
7267:
7209:
7141:
7090:
7057:
6952:
6909:
6529:10.1007/s00222-007-0054-1
6277:10.1007/978-3-642-57739-0
4409:Under these assumptions,
2756:, with Picard number 22.
2754:supersingular K3 surfaces
2432:{\displaystyle A/(\pm 1)}
403:coherent sheaf cohomology
7565:Introduction to M-theory
7259:WessâZuminoâWitten model
7201:HananyâWitten transition
6927:History of string theory
6757:Atti del Istituto Veneto
6498:Inventiones Mathematicae
6365:Experimental Mathematics
6265:Compact complex surfaces
5926:
5741:String compactifications
5367:{\displaystyle u^{2}=-2}
5308:{\displaystyle N^{1}(X)}
4779:{\displaystyle g\geq 63}
4682:{\displaystyle g\leq 13}
4601:; it can be viewed as a
4537:There is an irreducible
4445:) is said to have genus
3372:quasi-elliptic fibration
3173:{\displaystyle 20-\rho }
3068:{\displaystyle 20-\rho }
1311:{\displaystyle b_{3}(X)}
1271:{\displaystyle b_{1}(X)}
1242:. Thus the Betti number
869:{\displaystyle c_{i}(X)}
103:geometrically connected
7244:Vertex operator algebra
6944:String theory landscape
6697:Rudakov, A.N. (2001) ,
6655:PjateckiÄ-Ć apiro, I. I.
6557:Lectures on K3 surfaces
6304:(1983), "Surfaces K3",
5865:{\displaystyle g\geq 3}
5676:{\displaystyle \Delta }
5616:{\displaystyle \rho =2}
5528:{\displaystyle \Delta }
5485:. Namely, suppose that
5477:) determines the whole
5377:Case 2: Otherwise, let
4714:{\displaystyle g=18,20}
4594:{\displaystyle g\geq 2}
3363:{\displaystyle u^{2}=0}
3292:(a nodal cubic curve).
2022:symmetric bilinear form
1318:is also zero. Finally,
710:On the other hand, the
281:canonical singularities
80:is a compact connected
7542:AdS/CFT correspondence
7297:Exceptional Lie groups
7239:Superconformal algebra
7211:Conformal field theory
7082:MontonenâOlive duality
7034:Non-linear sigma model
5890:Ilya Piatetski-Shapiro
5866:
5840:
5787:
5677:
5617:
5587:
5555:
5529:
5515:. If the set of roots
5509:
5455:orthogonal complements
5443:
5368:
5309:
5273:
5235:
5153:
5112:
5083:
5002:
4971:
4940:
4905:
4874:
4836:
4780:
4750:
4715:
4683:
4645:
4595:
4565:
4528:
4491:
4464:gives a morphism from
4437:, which explains why (
4392:
4300:
4251:
4172:
4128:
4069:
4010:global Torelli theorem
4008:is not). However, the
3994:
3897:
3896:{\displaystyle N\to D}
3868:
3747:properly discontinuous
3739:
3710:
3666:
3595:
3535:
3443:
3404:
3364:
3331:
3286:
3259:
3220:
3174:
3148:
3122:
3069:
3043:
3020:
2987:
2949:
2904:
2866:
2827:
2783:
2734:
2714:
2690:
2627:
2545:
2513:
2486:The intersection of a
2462:
2433:
2380:
2337:
2301:
2228:
2154:
2123:
2058:
2014:
1931:
1867:
1805:
1616:
1574:
1510:
1443:
1354:
1312:
1272:
1236:
1186:
1097:
1019:
977:
909:
870:
831:
701:
549:
448:
395:
338:
207:, K3 surfaces are the
187:
60:
28:
8087:Differential geometry
7537:Holographic principle
7504:Type IIB supergravity
7499:Type IIA supergravity
7351:-form electrodynamics
6970:Bosonic string theory
6359:Brown, Gavin (2007),
6030:K3 database for Magma
5867:
5841:
5788:
5678:
5618:
5593:. In the first case,
5588:
5556:
5530:
5510:
5444:
5369:
5310:
5274:
5236:
5154:
5113:
5084:
5003:
4972:
4941:
4906:
4875:
4837:
4781:
4751:
4716:
4684:
4646:
4596:
4566:
4529:
4492:
4393:
4301:
4252:
4173:
4129:
4070:
3995:
3898:
3869:
3740:
3711:
3667:
3596:
3536:
3444:
3410:that contains a line
3405:
3365:
3332:
3287:
3285:{\displaystyle I_{1}}
3260:
3221:
3175:
3149:
3123:
3070:
3044:
3042:{\displaystyle \rho }
3021:
2988:
2950:
2905:
2867:
2865:{\displaystyle u^{2}}
2828:
2784:
2735:
2733:{\displaystyle \rho }
2715:
2713:{\displaystyle \rho }
2691:
2628:
2626:{\displaystyle \rho }
2546:
2514:
2463:
2439:can be embedded into
2434:
2381:
2338:
2307:is a smooth curve of
2302:
2229:
2155:
2153:{\displaystyle E_{8}}
2124:
2059:
2015:
1937:. For K3 surfaces in
1932:
1868:
1806:
1617:
1575:
1511:
1444:
1355:
1313:
1273:
1237:
1187:
1107:of cohomology groups
1098:
1020:
978:
910:
908:{\displaystyle K_{X}}
871:
832:
702:
550:
449:
396:
339:
213:hyperkÀhler manifolds
188:
76:, a complex analytic
31:
22:
7456:HoĆavaâWitten theory
7403:HyperkÀhler manifold
7091:Particles and fields
7039:Tachyon condensation
7024:Matrix string theory
6314:, pp. 217â229,
6022:Graded Ring Database
5850:
5821:
5768:
5764:Quartic surfaces in
5667:
5601:
5565:
5539:
5519:
5493:
5381:
5342:
5317:connected components
5283:
5245:
5183:
5134:
5093:
5035:
5015:K3 surface of genus
4981:
4950:
4919:
4884:
4853:
4790:
4764:
4729:
4693:
4667:
4624:
4579:
4544:
4509:
4472:
4468:to projective space
4344:
4261:
4182:
4138:
4094:
4019:
3911:
3907:to the complex line
3881:
3775:
3720:
3691:
3605:
3561:
3516:
3418:
3385:
3341:
3303:
3269:
3234:
3195:
3184:Elliptic K3 surfaces
3158:
3132:
3094:
3053:
3033:
2997:
2959:
2921:
2876:
2849:
2793:
2771:
2724:
2704:
2641:
2617:
2526:
2494:
2443:
2406:
2361:
2318:
2282:
2198:
2137:
2068:
2035:
1965:
1877:
1815:
1771:
1584:
1520:
1456:
1370:
1362:Euler characteristic
1322:
1286:
1246:
1196:
1111:
1035:
1030:exponential sequence
987:
919:
892:
844:
721:
712:RiemannâRoch theorem
573:
465:
409:
356:
303:
277:du Val singularities
209:CalabiâYau manifolds
125:
7494:Type I supergravity
7398:CalabiâYau manifold
7393:Ricci-flat manifold
7372:KaluzaâKlein theory
7113:RamondâRamond field
7019:String field theory
6675:1971IzMat...5..547P
6613:10.1112/jlms/jdu038
6521:2007InMat.169..519G
6422:10.24033/asens.1287
4539:coarse moduli space
3877:The period mapping
3487:linearly equivalent
2915:Hodge index theorem
2839:divisor class group
2835:intersection theory
2637:) is a subgroup of
1851:
1654:'s solution to the
1086:
333:
8077:Algebraic surfaces
7461:K-theory (physics)
7338:ADE classification
6975:Superstring theory
6552:Huybrechts, Daniel
6491:Gritsenko, V. A.;
6465:Enriques, Federigo
6438:Enriques, Federigo
6331:Bourguignon, J.-P.
5862:
5836:
5783:
5725:fundamental domain
5673:
5648:. Namely, let the
5631:Automorphism group
5613:
5583:
5551:
5525:
5505:
5489:has Picard number
5439:
5364:
5305:
5269:
5231:
5149:
5108:
5079:
5011:More generally, a
4998:
4967:
4936:
4901:
4870:
4832:
4776:
4746:
4711:
4679:
4641:
4591:
4561:
4524:
4487:
4388:
4296:
4247:
4168:
4124:
4065:
3990:
3893:
3864:
3745:is far from being
3735:
3706:
3662:
3601:to the K3 lattice
3591:
3531:
3439:
3400:
3360:
3327:
3282:
3255:
3216:
3190:elliptic fibration
3170:
3144:
3118:
3084:Viacheslav Nikulin
3065:
3039:
3016:
2983:
2945:
2900:
2862:
2823:
2779:
2748:of characteristic
2730:
2710:
2686:
2623:
2608:finitely generated
2563:The Picard lattice
2541:
2509:
2458:
2429:
2392:minimal resolution
2376:
2333:
2297:
2224:
2150:
2119:
2064:, or equivalently
2054:
2030:unimodular lattice
2010:
1927:
1863:
1837:
1801:
1612:
1580:, it follows that
1570:
1506:
1439:
1397:
1350:
1308:
1268:
1232:
1182:
1093:
1072:
1015:
973:
905:
866:
827:
697:
617:
545:
444:
391:
334:
319:
237:KacâMoody algebras
217:del Pezzo surfaces
183:
29:
8064:
8063:
7846:van Nieuwenhuizen
7382:Why 10 dimensions
7287:ChernâSimons form
7254:KacâMoody algebra
7234:Conformal algebra
7229:Conformal anomaly
7123:Magnetic monopole
7118:KalbâRamond field
6960:NambuâGoto action
6829:978-0-387-90330-9
6796:978-2-85629-253-2
6746:Severi, Francesco
6730:978-0-8218-3749-8
6401:Rapoport, Michael
6302:Beauville, Arnaud
6286:978-3-540-00832-3
5920:Mathieu group M24
5916:Mathieu moonshine
5807:Federigo Enriques
4419:Bertini's theorem
4331:ample line bundle
4004:is Hausdorff and
3850:
2872:is even for each
2593:Jean-Pierre Serre
1746:
1745:
1656:Calabi conjecture
1388:
764:
608:
352:) (the dimension
297:l-adic cohomology
229:abelian varieties
113:Kodaira dimension
105:algebraic surface
8099:
8082:Complex surfaces
7574:String theorists
7514:Lie superalgebra
7466:Twisted K-theory
7424:Spin(7)-manifold
7377:Compactification
7219:Virasoro algebra
7002:Heterotic string
6896:
6889:
6882:
6873:
6872:
6840:
6807:
6784:
6771:
6754:
6741:
6711:
6693:
6659:Ć afareviÄ, I. R.
6650:
6631:
6606:
6587:Verbitsky, Misha
6581:
6562:
6547:
6514:
6487:
6460:
6433:
6424:
6395:
6355:
6322:
6297:
6252:
6243:
6215:
6212:
6206:
6203:
6197:
6194:
6188:
6185:
6179:
6176:
6170:
6167:
6161:
6158:
6152:
6149:
6143:
6140:
6134:
6131:
6125:
6122:
6116:
6113:
6107:
6104:
6098:
6095:
6089:
6086:
6080:
6077:
6071:
6068:
6062:
6059:
6053:
6050:
6044:
6041:
6035:
6033:
6025:
6017:
6011:
6008:
6002:
5999:
5993:
5990:
5984:
5981:
5975:
5972:
5966:
5963:
5957:
5954:
5948:
5945:
5939:
5936:
5906:Enriques surface
5894:Michael Rapoport
5874:Francesco Severi
5871:
5869:
5868:
5863:
5845:
5843:
5842:
5837:
5835:
5834:
5829:
5793:were studied by
5792:
5790:
5789:
5784:
5782:
5781:
5776:
5754:Aspinwall (1996)
5682:
5680:
5679:
5674:
5646:commensurability
5622:
5620:
5619:
5614:
5592:
5590:
5589:
5584:
5560:
5558:
5557:
5552:
5534:
5532:
5531:
5526:
5514:
5512:
5511:
5506:
5448:
5446:
5445:
5440:
5426:
5425:
5373:
5371:
5370:
5365:
5354:
5353:
5314:
5312:
5311:
5306:
5295:
5294:
5278:
5276:
5275:
5270:
5240:
5238:
5237:
5232:
5230:
5195:
5194:
5158:
5156:
5155:
5150:
5148:
5147:
5142:
5126:is singular. (A
5117:
5115:
5114:
5109:
5107:
5106:
5101:
5088:
5086:
5085:
5080:
5063:
5062:
5047:
5046:
5007:
5005:
5004:
4999:
4997:
4996:
4991:
4990:
4976:
4974:
4973:
4968:
4966:
4965:
4960:
4959:
4945:
4943:
4942:
4937:
4935:
4934:
4929:
4928:
4910:
4908:
4907:
4902:
4900:
4899:
4894:
4893:
4879:
4877:
4876:
4871:
4869:
4868:
4863:
4862:
4841:
4839:
4838:
4833:
4785:
4783:
4782:
4777:
4755:
4753:
4752:
4747:
4745:
4744:
4739:
4738:
4720:
4718:
4717:
4712:
4688:
4686:
4685:
4680:
4653:quasi-projective
4650:
4648:
4647:
4642:
4640:
4639:
4634:
4633:
4600:
4598:
4597:
4592:
4570:
4568:
4567:
4562:
4560:
4559:
4554:
4553:
4533:
4531:
4530:
4525:
4523:
4522:
4517:
4496:
4494:
4493:
4488:
4486:
4485:
4480:
4397:
4395:
4394:
4389:
4372:
4371:
4356:
4355:
4305:
4303:
4302:
4297:
4292:
4291:
4273:
4272:
4256:
4254:
4253:
4248:
4243:
4229:
4228:
4213:
4212:
4194:
4193:
4177:
4175:
4174:
4169:
4164:
4150:
4149:
4133:
4131:
4130:
4125:
4120:
4106:
4105:
4074:
4072:
4071:
4066:
4052:
4029:
3999:
3997:
3996:
3991:
3989:
3972:
3958:
3957:
3942:
3941:
3923:
3922:
3902:
3900:
3899:
3894:
3873:
3871:
3870:
3865:
3851:
3843:
3825:
3824:
3809:
3744:
3742:
3741:
3736:
3715:
3713:
3712:
3707:
3686:orthogonal group
3671:
3669:
3668:
3663:
3661:
3660:
3645:
3644:
3623:
3622:
3600:
3598:
3597:
3592:
3587:
3573:
3572:
3540:
3538:
3537:
3532:
3530:
3529:
3524:
3494:Kobayashi metric
3448:
3446:
3445:
3440:
3438:
3437:
3432:
3409:
3407:
3406:
3401:
3399:
3398:
3393:
3369:
3367:
3366:
3361:
3353:
3352:
3336:
3334:
3333:
3328:
3291:
3289:
3288:
3283:
3281:
3280:
3264:
3262:
3261:
3256:
3225:
3223:
3222:
3217:
3215:
3214:
3209:
3179:
3177:
3176:
3171:
3153:
3151:
3150:
3145:
3127:
3125:
3124:
3119:
3074:
3072:
3071:
3066:
3048:
3046:
3045:
3040:
3025:
3023:
3022:
3017:
3009:
3008:
2992:
2990:
2989:
2984:
2954:
2952:
2951:
2946:
2909:
2907:
2906:
2901:
2871:
2869:
2868:
2863:
2861:
2860:
2832:
2830:
2829:
2824:
2819:
2805:
2804:
2788:
2786:
2785:
2780:
2778:
2739:
2737:
2736:
2731:
2719:
2717:
2716:
2711:
2695:
2693:
2692:
2687:
2685:
2684:
2679:
2667:
2653:
2652:
2632:
2630:
2629:
2624:
2550:
2548:
2547:
2542:
2540:
2539:
2534:
2518:
2516:
2515:
2510:
2508:
2507:
2502:
2467:
2465:
2464:
2459:
2457:
2456:
2451:
2438:
2436:
2435:
2430:
2416:
2385:
2383:
2382:
2377:
2342:
2340:
2339:
2334:
2332:
2331:
2326:
2306:
2304:
2303:
2298:
2296:
2295:
2290:
2264:projective plane
2243:Michael Freedman
2233:
2231:
2230:
2225:
2223:
2222:
2210:
2209:
2159:
2157:
2156:
2151:
2149:
2148:
2128:
2126:
2125:
2120:
2118:
2117:
2102:
2101:
2080:
2079:
2063:
2061:
2060:
2055:
2053:
2052:
2019:
2017:
2016:
2011:
2009:
2008:
2003:
1991:
1977:
1976:
1946:Igor Shafarevich
1936:
1934:
1933:
1928:
1926:
1925:
1920:
1908:
1907:
1889:
1888:
1872:
1870:
1869:
1864:
1862:
1850:
1845:
1827:
1826:
1810:
1808:
1807:
1802:
1797:
1783:
1782:
1675:
1674:
1637:Kunihiko Kodaira
1621:
1619:
1618:
1613:
1596:
1595:
1579:
1577:
1576:
1571:
1554:
1553:
1532:
1531:
1515:
1513:
1512:
1507:
1490:
1489:
1468:
1467:
1448:
1446:
1445:
1440:
1426:
1425:
1416:
1415:
1396:
1359:
1357:
1356:
1351:
1334:
1333:
1317:
1315:
1314:
1309:
1298:
1297:
1280:Poincaré duality
1278:is zero, and by
1277:
1275:
1274:
1269:
1258:
1257:
1241:
1239:
1238:
1233:
1222:
1208:
1207:
1191:
1189:
1188:
1183:
1178:
1177:
1159:
1158:
1143:
1129:
1128:
1102:
1100:
1099:
1094:
1085:
1080:
1068:
1067:
1055:
1054:
1049:
1024:
1022:
1021:
1016:
999:
998:
983:is zero, and so
982:
980:
979:
974:
963:
962:
944:
943:
931:
930:
914:
912:
911:
906:
904:
903:
875:
873:
872:
867:
856:
855:
836:
834:
833:
828:
823:
819:
809:
808:
796:
795:
780:
779:
765:
757:
749:
748:
743:
742:
706:
704:
703:
698:
669:
668:
663:
662:
646:
645:
636:
635:
616:
601:
600:
595:
594:
554:
552:
551:
546:
535:
534:
516:
515:
500:
499:
494:
493:
477:
476:
453:
451:
450:
445:
440:
439:
421:
420:
400:
398:
397:
392:
387:
386:
368:
367:
343:
341:
340:
335:
332:
327:
315:
314:
262:simply connected
192:
190:
189:
184:
176:
175:
163:
162:
150:
149:
137:
136:
86:canonical bundle
82:complex manifold
68:
8107:
8106:
8102:
8101:
8100:
8098:
8097:
8096:
8067:
8066:
8065:
8060:
7569:
7546:
7523:
7470:
7418:
7388:KĂ€hler manifold
7355:
7332:
7325:
7318:
7311:
7304:
7263:
7224:Mirror symmetry
7205:
7191:Brane cosmology
7137:
7086:
7053:
7009:N=2 superstring
6995:Type IIB string
6990:Type IIA string
6965:Polyakov action
6948:
6905:
6900:
6847:
6830:
6820:Springer-Verlag
6797:
6782:
6752:
6731:
6571:
6560:
6399:Burns, Daniel;
6287:
6228:Aspinwall, Paul
6223:
6218:
6213:
6209:
6204:
6200:
6195:
6191:
6186:
6182:
6177:
6173:
6168:
6164:
6159:
6155:
6150:
6146:
6141:
6137:
6132:
6128:
6123:
6119:
6114:
6110:
6105:
6101:
6096:
6092:
6087:
6083:
6078:
6074:
6069:
6065:
6060:
6056:
6051:
6047:
6042:
6038:
6027:
6019:
6018:
6014:
6009:
6005:
6000:
5996:
5991:
5987:
5982:
5978:
5973:
5969:
5964:
5960:
5955:
5951:
5946:
5942:
5937:
5933:
5929:
5911:Tate conjecture
5902:
5851:
5848:
5847:
5830:
5825:
5824:
5822:
5819:
5818:
5803:Friedrich Schur
5777:
5772:
5771:
5769:
5766:
5765:
5762:
5750:
5746:
5733:
5689:normal subgroup
5668:
5665:
5664:
5633:
5602:
5599:
5598:
5566:
5563:
5562:
5540:
5537:
5536:
5520:
5517:
5516:
5494:
5491:
5490:
5421:
5417:
5382:
5379:
5378:
5349:
5345:
5343:
5340:
5339:
5290:
5286:
5284:
5281:
5280:
5246:
5243:
5242:
5226:
5190:
5186:
5184:
5181:
5180:
5173:
5143:
5138:
5137:
5135:
5132:
5131:
5102:
5097:
5096:
5094:
5091:
5090:
5058:
5054:
5042:
5038:
5036:
5033:
5032:
5013:quasi-polarized
4992:
4986:
4985:
4984:
4982:
4979:
4978:
4961:
4955:
4954:
4953:
4951:
4948:
4947:
4930:
4924:
4923:
4922:
4920:
4917:
4916:
4895:
4889:
4888:
4887:
4885:
4882:
4881:
4864:
4858:
4857:
4856:
4854:
4851:
4850:
4791:
4788:
4787:
4765:
4762:
4761:
4740:
4734:
4733:
4732:
4730:
4727:
4726:
4694:
4691:
4690:
4668:
4665:
4664:
4635:
4629:
4628:
4627:
4625:
4622:
4621:
4607:Shimura variety
4580:
4577:
4576:
4555:
4549:
4548:
4547:
4545:
4542:
4541:
4518:
4513:
4512:
4510:
4507:
4506:
4481:
4476:
4475:
4473:
4470:
4469:
4367:
4363:
4351:
4347:
4345:
4342:
4341:
4312:
4287:
4283:
4268:
4264:
4262:
4259:
4258:
4239:
4224:
4220:
4208:
4204:
4189:
4185:
4183:
4180:
4179:
4160:
4145:
4141:
4139:
4136:
4135:
4116:
4101:
4097:
4095:
4092:
4091:
4048:
4025:
4020:
4017:
4016:
3985:
3968:
3953:
3949:
3937:
3933:
3918:
3914:
3912:
3909:
3908:
3882:
3879:
3878:
3842:
3820:
3816:
3805:
3776:
3773:
3772:
3766:Torelli theorem
3762:Hodge structure
3721:
3718:
3717:
3692:
3689:
3688:
3653:
3649:
3637:
3633:
3618:
3614:
3606:
3603:
3602:
3583:
3568:
3564:
3562:
3559:
3558:
3547:
3525:
3520:
3519:
3517:
3514:
3513:
3475:Fedor Bogomolov
3459:
3433:
3428:
3427:
3419:
3416:
3415:
3394:
3389:
3388:
3386:
3383:
3382:
3348:
3344:
3342:
3339:
3338:
3304:
3301:
3300:
3276:
3272:
3270:
3267:
3266:
3235:
3232:
3231:
3228:rational curves
3210:
3205:
3204:
3196:
3193:
3192:
3186:
3159:
3156:
3155:
3133:
3130:
3129:
3095:
3092:
3091:
3054:
3051:
3050:
3034:
3031:
3030:
3004:
3000:
2998:
2995:
2994:
2960:
2957:
2956:
2922:
2919:
2918:
2877:
2874:
2873:
2856:
2852:
2850:
2847:
2846:
2815:
2800:
2796:
2794:
2791:
2790:
2774:
2772:
2769:
2768:
2725:
2722:
2721:
2705:
2702:
2701:
2680:
2675:
2674:
2663:
2648:
2644:
2642:
2639:
2638:
2618:
2615:
2614:
2565:
2535:
2530:
2529:
2527:
2524:
2523:
2503:
2498:
2497:
2495:
2492:
2491:
2490:and a cubic in
2452:
2447:
2446:
2444:
2441:
2440:
2412:
2407:
2404:
2403:
2362:
2359:
2358:
2352:abelian variety
2327:
2322:
2321:
2319:
2316:
2315:
2291:
2286:
2285:
2283:
2280:
2279:
2252:
2218:
2214:
2205:
2201:
2199:
2196:
2195:
2172:11/8 conjecture
2168:Yukio Matsumoto
2144:
2140:
2138:
2135:
2134:
2110:
2106:
2094:
2090:
2075:
2071:
2069:
2066:
2065:
2042:
2038:
2036:
2033:
2032:
2004:
1999:
1998:
1987:
1972:
1968:
1966:
1963:
1962:
1921:
1916:
1915:
1903:
1899:
1884:
1880:
1878:
1875:
1874:
1858:
1846:
1841:
1822:
1818:
1816:
1813:
1812:
1793:
1778:
1774:
1772:
1769:
1768:
1765:Hodge structure
1757:Hodge structure
1628:
1591:
1587:
1585:
1582:
1581:
1549:
1545:
1527:
1523:
1521:
1518:
1517:
1485:
1481:
1463:
1459:
1457:
1454:
1453:
1421:
1417:
1411:
1407:
1392:
1371:
1368:
1367:
1329:
1325:
1323:
1320:
1319:
1293:
1289:
1287:
1284:
1283:
1253:
1249:
1247:
1244:
1243:
1218:
1203:
1199:
1197:
1194:
1193:
1173:
1169:
1154:
1150:
1139:
1124:
1120:
1112:
1109:
1108:
1081:
1076:
1063:
1059:
1050:
1045:
1044:
1036:
1033:
1032:
994:
990:
988:
985:
984:
958:
954:
939:
935:
926:
922:
920:
917:
916:
899:
895:
893:
890:
889:
851:
847:
845:
842:
841:
804:
800:
791:
787:
775:
771:
770:
766:
756:
744:
738:
737:
736:
722:
719:
718:
664:
658:
657:
656:
641:
637:
631:
627:
612:
596:
590:
589:
588:
574:
571:
570:
530:
526:
511:
507:
495:
489:
488:
487:
472:
468:
466:
463:
462:
435:
431:
416:
412:
410:
407:
406:
382:
378:
363:
359:
357:
354:
353:
328:
323:
310:
306:
304:
301:
300:
289:
257:
241:mirror symmetry
171:
167:
158:
154:
145:
141:
132:
128:
126:
123:
122:
117:quartic surface
70:
62:
35:
34:
17:
12:
11:
5:
8105:
8095:
8094:
8089:
8084:
8079:
8062:
8061:
8059:
8058:
8053:
8048:
8043:
8038:
8033:
8028:
8023:
8018:
8013:
8008:
8003:
7998:
7993:
7988:
7983:
7978:
7973:
7968:
7963:
7958:
7953:
7948:
7943:
7938:
7933:
7928:
7923:
7918:
7913:
7908:
7903:
7898:
7896:Randjbar-Daemi
7893:
7888:
7883:
7878:
7873:
7868:
7863:
7858:
7853:
7848:
7843:
7838:
7833:
7828:
7823:
7818:
7813:
7808:
7803:
7798:
7793:
7788:
7783:
7778:
7773:
7768:
7763:
7758:
7753:
7748:
7743:
7738:
7733:
7728:
7723:
7718:
7713:
7708:
7703:
7698:
7693:
7688:
7683:
7678:
7673:
7668:
7663:
7658:
7653:
7648:
7643:
7638:
7633:
7628:
7623:
7618:
7613:
7608:
7603:
7598:
7593:
7588:
7583:
7577:
7575:
7571:
7570:
7568:
7567:
7562:
7556:
7554:
7548:
7547:
7545:
7544:
7539:
7533:
7531:
7525:
7524:
7522:
7521:
7519:Lie supergroup
7516:
7511:
7506:
7501:
7496:
7491:
7486:
7480:
7478:
7472:
7471:
7469:
7468:
7463:
7458:
7453:
7448:
7443:
7438:
7433:
7428:
7427:
7426:
7421:
7416:
7412:
7411:
7410:
7400:
7390:
7385:
7379:
7374:
7369:
7363:
7361:
7357:
7356:
7354:
7353:
7345:
7340:
7335:
7330:
7323:
7316:
7309:
7302:
7294:
7289:
7284:
7279:
7273:
7271:
7265:
7264:
7262:
7261:
7256:
7251:
7246:
7241:
7236:
7231:
7226:
7221:
7215:
7213:
7207:
7206:
7204:
7203:
7198:
7196:Quiver diagram
7193:
7188:
7183:
7178:
7173:
7168:
7163:
7158:
7153:
7147:
7145:
7139:
7138:
7136:
7135:
7130:
7125:
7120:
7115:
7110:
7105:
7100:
7094:
7092:
7088:
7087:
7085:
7084:
7079:
7074:
7069:
7063:
7061:
7059:String duality
7055:
7054:
7052:
7051:
7046:
7041:
7036:
7031:
7026:
7021:
7016:
7011:
7006:
7005:
7004:
6999:
6998:
6997:
6992:
6985:Type II string
6982:
6972:
6967:
6962:
6956:
6954:
6950:
6949:
6947:
6946:
6941:
6940:
6939:
6934:
6924:
6922:Cosmic strings
6919:
6913:
6911:
6907:
6906:
6899:
6898:
6891:
6884:
6876:
6870:
6869:
6863:
6854:
6846:
6845:External links
6843:
6842:
6841:
6828:
6808:
6795:
6776:Voisin, Claire
6772:
6742:
6729:
6712:
6694:
6669:(3): 547â588,
6651:
6636:Mukai, Shigeru
6632:
6597:(2): 436â450,
6582:
6570:978-1107153042
6569:
6548:
6505:(3): 519â567,
6488:
6461:
6434:
6415:(2): 235â273,
6396:
6356:
6323:
6298:
6285:
6257:Barth, Wolf P.
6253:
6241:hep-th/9611137
6222:
6219:
6217:
6216:
6207:
6198:
6189:
6180:
6171:
6162:
6153:
6144:
6135:
6126:
6117:
6108:
6099:
6090:
6081:
6072:
6063:
6054:
6045:
6036:
6012:
6003:
5994:
5985:
5976:
5967:
5958:
5949:
5940:
5930:
5928:
5925:
5924:
5923:
5913:
5908:
5901:
5898:
5861:
5858:
5855:
5833:
5828:
5780:
5775:
5761:
5758:
5748:
5744:
5737:string duality
5732:
5729:
5672:
5632:
5629:
5612:
5609:
5606:
5582:
5579:
5576:
5573:
5570:
5550:
5547:
5544:
5524:
5504:
5501:
5498:
5479:cone of curves
5438:
5435:
5432:
5429:
5424:
5420:
5416:
5413:
5410:
5407:
5404:
5401:
5398:
5395:
5392:
5389:
5386:
5363:
5360:
5357:
5352:
5348:
5304:
5301:
5298:
5293:
5289:
5268:
5265:
5262:
5259:
5256:
5253:
5250:
5241:has signature
5229:
5225:
5222:
5219:
5216:
5213:
5210:
5207:
5204:
5201:
5198:
5193:
5189:
5172:
5169:
5146:
5141:
5105:
5100:
5078:
5075:
5072:
5069:
5066:
5061:
5057:
5053:
5050:
5045:
5041:
4995:
4989:
4964:
4958:
4933:
4927:
4898:
4892:
4867:
4861:
4831:
4828:
4825:
4822:
4819:
4816:
4813:
4810:
4807:
4804:
4801:
4798:
4795:
4775:
4772:
4769:
4743:
4737:
4710:
4707:
4704:
4701:
4698:
4678:
4675:
4672:
4638:
4632:
4609:for the group
4590:
4587:
4584:
4558:
4552:
4521:
4516:
4484:
4479:
4456:has dimension
4415:basepoint-free
4387:
4384:
4381:
4378:
4375:
4370:
4366:
4362:
4359:
4354:
4350:
4311:
4308:
4295:
4290:
4286:
4282:
4279:
4276:
4271:
4267:
4246:
4242:
4238:
4235:
4232:
4227:
4223:
4219:
4216:
4211:
4207:
4203:
4200:
4197:
4192:
4188:
4167:
4163:
4159:
4156:
4153:
4148:
4144:
4123:
4119:
4115:
4112:
4109:
4104:
4100:
4088:Hodge isometry
4076:
4075:
4064:
4061:
4058:
4055:
4051:
4047:
4044:
4041:
4038:
4035:
4032:
4028:
4024:
3988:
3984:
3981:
3978:
3975:
3971:
3967:
3964:
3961:
3956:
3952:
3948:
3945:
3940:
3936:
3932:
3929:
3926:
3921:
3917:
3892:
3889:
3886:
3875:
3874:
3863:
3860:
3857:
3854:
3849:
3846:
3841:
3838:
3834:
3831:
3828:
3823:
3819:
3815:
3812:
3808:
3804:
3801:
3798:
3795:
3792:
3789:
3786:
3783:
3780:
3758:period mapping
3734:
3731:
3728:
3725:
3705:
3702:
3699:
3696:
3659:
3656:
3652:
3648:
3643:
3640:
3636:
3632:
3629:
3626:
3621:
3617:
3613:
3610:
3590:
3586:
3582:
3579:
3576:
3571:
3567:
3546:
3545:The period map
3543:
3528:
3523:
3458:
3455:
3436:
3431:
3426:
3423:
3397:
3392:
3359:
3356:
3351:
3347:
3326:
3323:
3320:
3317:
3314:
3311:
3308:
3279:
3275:
3254:
3251:
3248:
3245:
3242:
3239:
3213:
3208:
3203:
3200:
3185:
3182:
3169:
3166:
3163:
3143:
3140:
3137:
3117:
3114:
3111:
3108:
3105:
3102:
3099:
3088:David Morrison
3064:
3061:
3058:
3049:has dimension
3038:
3015:
3012:
3007:
3003:
2982:
2979:
2976:
2973:
2970:
2967:
2964:
2944:
2941:
2938:
2935:
2932:
2929:
2926:
2899:
2896:
2893:
2890:
2887:
2884:
2881:
2859:
2855:
2822:
2818:
2814:
2811:
2808:
2803:
2799:
2777:
2761:Picard lattice
2729:
2709:
2683:
2678:
2673:
2670:
2666:
2662:
2659:
2656:
2651:
2647:
2622:
2564:
2561:
2560:
2559:
2552:
2538:
2533:
2520:
2506:
2501:
2484:
2473:
2455:
2450:
2428:
2425:
2422:
2419:
2415:
2411:
2375:
2372:
2369:
2366:
2357:by the action
2348:Kummer surface
2344:
2330:
2325:
2312:
2294:
2289:
2251:
2248:
2247:
2246:
2235:
2221:
2217:
2213:
2208:
2204:
2165:
2147:
2143:
2116:
2113:
2109:
2105:
2100:
2097:
2093:
2089:
2086:
2083:
2078:
2074:
2051:
2048:
2045:
2041:
2007:
2002:
1997:
1994:
1990:
1986:
1983:
1980:
1975:
1971:
1951:
1950:
1949:
1939:characteristic
1924:
1919:
1914:
1911:
1906:
1902:
1898:
1895:
1892:
1887:
1883:
1861:
1857:
1854:
1849:
1844:
1840:
1836:
1833:
1830:
1825:
1821:
1800:
1796:
1792:
1789:
1786:
1781:
1777:
1753:Jacobian ideal
1749:
1748:
1747:
1744:
1743:
1741:
1739:
1736:
1734:
1731:
1730:
1728:
1725:
1723:
1720:
1717:
1716:
1713:
1711:
1708:
1706:
1702:
1701:
1699:
1696:
1694:
1691:
1688:
1687:
1685:
1683:
1680:
1678:
1663:
1662:KĂ€hler metric.
1652:Shing-Tung Yau
1640:
1627:
1624:
1611:
1608:
1605:
1602:
1599:
1594:
1590:
1569:
1566:
1563:
1560:
1557:
1552:
1548:
1544:
1541:
1538:
1535:
1530:
1526:
1505:
1502:
1499:
1496:
1493:
1488:
1484:
1480:
1477:
1474:
1471:
1466:
1462:
1450:
1449:
1438:
1435:
1432:
1429:
1424:
1420:
1414:
1410:
1406:
1403:
1400:
1395:
1391:
1387:
1384:
1381:
1378:
1375:
1349:
1346:
1343:
1340:
1337:
1332:
1328:
1307:
1304:
1301:
1296:
1292:
1267:
1264:
1261:
1256:
1252:
1231:
1228:
1225:
1221:
1217:
1214:
1211:
1206:
1202:
1181:
1176:
1172:
1168:
1165:
1162:
1157:
1153:
1149:
1146:
1142:
1138:
1135:
1132:
1127:
1123:
1119:
1116:
1105:exact sequence
1092:
1089:
1084:
1079:
1075:
1071:
1066:
1062:
1058:
1053:
1048:
1043:
1040:
1014:
1011:
1008:
1005:
1002:
997:
993:
972:
969:
966:
961:
957:
953:
950:
947:
942:
938:
934:
929:
925:
902:
898:
886:tangent bundle
865:
862:
859:
854:
850:
838:
837:
826:
822:
818:
815:
812:
807:
803:
799:
794:
790:
786:
783:
778:
774:
769:
763:
760:
755:
752:
747:
741:
735:
732:
729:
726:
708:
707:
696:
693:
690:
687:
684:
681:
678:
675:
672:
667:
661:
655:
652:
649:
644:
640:
634:
630:
626:
623:
620:
615:
611:
607:
604:
599:
593:
587:
584:
581:
578:
556:
555:
544:
541:
538:
533:
529:
525:
522:
519:
514:
510:
506:
503:
498:
492:
486:
483:
480:
475:
471:
454:) is zero. By
443:
438:
434:
430:
427:
424:
419:
415:
390:
385:
381:
377:
374:
371:
366:
362:
331:
326:
322:
318:
313:
309:
288:
285:
256:
253:
211:(and also the
194:
193:
182:
179:
174:
170:
166:
161:
157:
153:
148:
144:
140:
135:
131:
30:
15:
9:
6:
4:
3:
2:
8104:
8093:
8092:String theory
8090:
8088:
8085:
8083:
8080:
8078:
8075:
8074:
8072:
8057:
8054:
8052:
8049:
8047:
8044:
8042:
8041:Zamolodchikov
8039:
8037:
8036:Zamolodchikov
8034:
8032:
8029:
8027:
8024:
8022:
8019:
8017:
8014:
8012:
8009:
8007:
8004:
8002:
7999:
7997:
7994:
7992:
7989:
7987:
7984:
7982:
7979:
7977:
7974:
7972:
7969:
7967:
7964:
7962:
7959:
7957:
7954:
7952:
7949:
7947:
7944:
7942:
7939:
7937:
7934:
7932:
7929:
7927:
7924:
7922:
7919:
7917:
7914:
7912:
7909:
7907:
7904:
7902:
7899:
7897:
7894:
7892:
7889:
7887:
7884:
7882:
7879:
7877:
7874:
7872:
7869:
7867:
7864:
7862:
7859:
7857:
7854:
7852:
7849:
7847:
7844:
7842:
7839:
7837:
7834:
7832:
7829:
7827:
7824:
7822:
7819:
7817:
7814:
7812:
7809:
7807:
7804:
7802:
7799:
7797:
7794:
7792:
7789:
7787:
7784:
7782:
7779:
7777:
7774:
7772:
7769:
7767:
7764:
7762:
7759:
7757:
7754:
7752:
7749:
7747:
7744:
7742:
7739:
7737:
7734:
7732:
7729:
7727:
7724:
7722:
7719:
7717:
7714:
7712:
7709:
7707:
7704:
7702:
7699:
7697:
7694:
7692:
7689:
7687:
7684:
7682:
7679:
7677:
7674:
7672:
7669:
7667:
7664:
7662:
7659:
7657:
7654:
7652:
7649:
7647:
7644:
7642:
7639:
7637:
7634:
7632:
7629:
7627:
7624:
7622:
7619:
7617:
7614:
7612:
7609:
7607:
7604:
7602:
7599:
7597:
7594:
7592:
7589:
7587:
7584:
7582:
7579:
7578:
7576:
7572:
7566:
7563:
7561:
7560:Matrix theory
7558:
7557:
7555:
7553:
7549:
7543:
7540:
7538:
7535:
7534:
7532:
7530:
7526:
7520:
7517:
7515:
7512:
7510:
7507:
7505:
7502:
7500:
7497:
7495:
7492:
7490:
7487:
7485:
7482:
7481:
7479:
7477:
7476:Supersymmetry
7473:
7467:
7464:
7462:
7459:
7457:
7454:
7452:
7449:
7447:
7444:
7442:
7439:
7437:
7434:
7432:
7429:
7425:
7422:
7420:
7413:
7409:
7406:
7405:
7404:
7401:
7399:
7396:
7395:
7394:
7391:
7389:
7386:
7383:
7380:
7378:
7375:
7373:
7370:
7368:
7365:
7364:
7362:
7358:
7352:
7350:
7346:
7344:
7341:
7339:
7336:
7333:
7326:
7319:
7312:
7305:
7298:
7295:
7293:
7290:
7288:
7285:
7283:
7280:
7278:
7275:
7274:
7272:
7270:
7266:
7260:
7257:
7255:
7252:
7250:
7247:
7245:
7242:
7240:
7237:
7235:
7232:
7230:
7227:
7225:
7222:
7220:
7217:
7216:
7214:
7212:
7208:
7202:
7199:
7197:
7194:
7192:
7189:
7187:
7184:
7182:
7179:
7177:
7174:
7172:
7169:
7167:
7164:
7162:
7159:
7157:
7154:
7152:
7149:
7148:
7146:
7144:
7140:
7134:
7131:
7129:
7128:Dual graviton
7126:
7124:
7121:
7119:
7116:
7114:
7111:
7109:
7106:
7104:
7101:
7099:
7096:
7095:
7093:
7089:
7083:
7080:
7078:
7075:
7073:
7070:
7068:
7065:
7064:
7062:
7060:
7056:
7050:
7047:
7045:
7044:RNS formalism
7042:
7040:
7037:
7035:
7032:
7030:
7027:
7025:
7022:
7020:
7017:
7015:
7012:
7010:
7007:
7003:
7000:
6996:
6993:
6991:
6988:
6987:
6986:
6983:
6981:
6980:Type I string
6978:
6977:
6976:
6973:
6971:
6968:
6966:
6963:
6961:
6958:
6957:
6955:
6951:
6945:
6942:
6938:
6935:
6933:
6930:
6929:
6928:
6925:
6923:
6920:
6918:
6915:
6914:
6912:
6908:
6904:
6903:String theory
6897:
6892:
6890:
6885:
6883:
6878:
6877:
6874:
6867:
6864:
6862:
6858:
6855:
6852:
6849:
6848:
6839:
6835:
6831:
6825:
6821:
6817:
6813:
6809:
6806:
6802:
6798:
6792:
6788:
6781:
6777:
6773:
6770:
6766:
6762:
6758:
6751:
6747:
6743:
6740:
6736:
6732:
6726:
6722:
6718:
6713:
6710:
6706:
6705:
6700:
6695:
6692:
6688:
6684:
6680:
6676:
6672:
6668:
6664:
6660:
6656:
6652:
6649:
6645:
6641:
6637:
6633:
6630:
6626:
6622:
6618:
6614:
6610:
6605:
6600:
6596:
6592:
6588:
6583:
6580:
6576:
6572:
6566:
6559:
6558:
6553:
6549:
6546:
6542:
6538:
6534:
6530:
6526:
6522:
6518:
6513:
6508:
6504:
6500:
6499:
6494:
6489:
6486:
6482:
6478:
6474:
6470:
6466:
6462:
6459:
6455:
6451:
6447:
6443:
6439:
6435:
6432:
6428:
6423:
6418:
6414:
6410:
6406:
6402:
6397:
6394:
6390:
6386:
6382:
6378:
6374:
6370:
6366:
6362:
6357:
6354:
6350:
6346:
6342:
6341:
6336:
6332:
6328:
6327:Beauville, A.
6324:
6321:
6317:
6313:
6309:
6308:
6303:
6299:
6296:
6292:
6288:
6282:
6278:
6274:
6270:
6266:
6262:
6258:
6254:
6251:
6247:
6242:
6237:
6233:
6229:
6225:
6224:
6211:
6202:
6193:
6184:
6175:
6166:
6157:
6148:
6139:
6130:
6121:
6112:
6103:
6094:
6085:
6076:
6067:
6058:
6049:
6040:
6032:
6031:
6024:
6023:
6016:
6007:
5998:
5989:
5980:
5971:
5962:
5953:
5944:
5935:
5931:
5921:
5917:
5914:
5912:
5909:
5907:
5904:
5903:
5897:
5895:
5891:
5886:
5881:
5879:
5875:
5859:
5856:
5853:
5831:
5816:
5812:
5808:
5804:
5800:
5799:Arthur Cayley
5796:
5778:
5757:
5755:
5742:
5738:
5728:
5726:
5722:
5718:
5714:
5710:
5706:
5702:
5698:
5694:
5690:
5686:
5662:
5658:
5654:
5651:
5647:
5643:
5639:
5628:
5626:
5610:
5607:
5604:
5596:
5580:
5577:
5574:
5571:
5568:
5545:
5542:
5502:
5499:
5496:
5488:
5484:
5480:
5476:
5472:
5467:
5465:
5461:
5456:
5452:
5449:, the set of
5433:
5430:
5427:
5422:
5418:
5414:
5408:
5402:
5399:
5396:
5393:
5387:
5375:
5361:
5358:
5355:
5350:
5346:
5337:
5333:
5328:
5326:
5322:
5321:positive cone
5318:
5299:
5291:
5287:
5263:
5260:
5257:
5254:
5251:
5223:
5217:
5211:
5208:
5205:
5199:
5191:
5187:
5178:
5168:
5166:
5162:
5144:
5129:
5125:
5121:
5103:
5076:
5073:
5070:
5067:
5064:
5059:
5051:
5043:
5039:
5030:
5026:
5022:
5018:
5014:
5009:
4993:
4962:
4931:
4914:
4896:
4865:
4847:
4845:
4844:Voisin (2008)
4829:
4826:
4823:
4820:
4817:
4814:
4811:
4808:
4805:
4802:
4799:
4796:
4793:
4773:
4770:
4767:
4759:
4741:
4724:
4708:
4705:
4702:
4699:
4696:
4676:
4673:
4670:
4662:
4658:
4657:Shigeru Mukai
4654:
4636:
4619:
4615:
4613:
4608:
4604:
4588:
4585:
4582:
4574:
4556:
4540:
4535:
4519:
4504:
4500:
4482:
4467:
4463:
4459:
4455:
4450:
4448:
4444:
4440:
4436:
4432:
4428:
4427:linear system
4424:
4420:
4416:
4412:
4407:
4405:
4401:
4385:
4382:
4379:
4376:
4373:
4368:
4360:
4352:
4348:
4339:
4335:
4332:
4328:
4325:
4321:
4317:
4307:
4288:
4280:
4277:
4269:
4265:
4236:
4233:
4225:
4221:
4217:
4209:
4201:
4198:
4190:
4186:
4157:
4154:
4146:
4142:
4113:
4110:
4102:
4098:
4089:
4085:
4081:
4053:
4049:
4045:
4030:
4026:
4022:
4015:
4014:
4013:
4011:
4007:
4003:
3982:
3976:
3965:
3962:
3954:
3950:
3946:
3938:
3930:
3927:
3919:
3915:
3906:
3890:
3884:
3861:
3855:
3852:
3844:
3839:
3836:
3832:
3829:
3826:
3821:
3817:
3813:
3802:
3793:
3790:
3787:
3781:
3778:
3771:
3770:
3769:
3767:
3763:
3759:
3754:
3752:
3748:
3723:
3694:
3687:
3683:
3679:
3675:
3657:
3654:
3650:
3646:
3641:
3638:
3630:
3627:
3619:
3615:
3611:
3580:
3577:
3569:
3565:
3556:
3552:
3542:
3526:
3511:
3507:
3503:
3499:
3495:
3490:
3488:
3484:
3480:
3479:David Mumford
3476:
3472:
3468:
3464:
3454:
3452:
3434:
3421:
3413:
3395:
3380:
3375:
3373:
3357:
3354:
3349:
3345:
3321:
3315:
3312:
3309:
3306:
3298:
3293:
3277:
3273:
3252:
3249:
3243:
3237:
3229:
3211:
3198:
3191:
3181:
3167:
3164:
3161:
3141:
3138:
3135:
3112:
3109:
3106:
3103:
3100:
3089:
3085:
3080:
3078:
3077:moduli spaces
3062:
3059:
3056:
3036:
3027:
3013:
3010:
3005:
3001:
2977:
2971:
2968:
2965:
2962:
2939:
2936:
2933:
2930:
2927:
2916:
2911:
2894:
2888:
2885:
2882:
2879:
2857:
2853:
2844:
2840:
2836:
2812:
2809:
2801:
2797:
2766:
2762:
2757:
2755:
2751:
2747:
2743:
2727:
2707:
2699:
2681:
2671:
2660:
2657:
2649:
2645:
2636:
2620:
2613:
2612:Picard number
2609:
2605:
2600:
2598:
2594:
2590:
2586:
2582:
2578:
2574:
2570:
2557:
2553:
2536:
2521:
2504:
2489:
2485:
2482:
2478:
2474:
2471:
2453:
2423:
2420:
2413:
2409:
2401:
2397:
2393:
2389:
2373:
2370:
2364:
2356:
2353:
2349:
2345:
2328:
2313:
2310:
2292:
2277:
2274:of a general
2273:
2269:
2265:
2261:
2258:
2254:
2253:
2244:
2240:
2236:
2219:
2215:
2211:
2206:
2202:
2193:
2192:connected sum
2189:
2185:
2181:
2177:
2173:
2169:
2166:
2163:
2145:
2141:
2132:
2114:
2111:
2107:
2103:
2098:
2095:
2087:
2084:
2076:
2072:
2049:
2046:
2043:
2039:
2031:
2027:
2023:
2005:
1995:
1984:
1981:
1973:
1969:
1960:
1956:
1952:
1947:
1943:
1940:
1922:
1912:
1904:
1896:
1893:
1885:
1881:
1855:
1847:
1842:
1834:
1831:
1823:
1819:
1790:
1787:
1779:
1775:
1766:
1762:
1758:
1754:
1750:
1742:
1740:
1737:
1735:
1733:
1732:
1729:
1726:
1724:
1721:
1719:
1718:
1714:
1712:
1709:
1707:
1704:
1703:
1700:
1697:
1695:
1692:
1690:
1689:
1686:
1684:
1681:
1679:
1677:
1676:
1673:
1672:
1671:
1670:
1668:
1667:Hodge numbers
1664:
1661:
1657:
1653:
1649:
1645:
1644:KĂ€hler metric
1641:
1638:
1634:
1633:diffeomorphic
1630:
1629:
1623:
1609:
1606:
1600:
1592:
1588:
1567:
1564:
1558:
1550:
1546:
1542:
1536:
1528:
1524:
1503:
1500:
1494:
1486:
1482:
1478:
1472:
1464:
1460:
1436:
1430:
1422:
1418:
1412:
1404:
1401:
1393:
1389:
1385:
1379:
1373:
1366:
1365:
1364:
1363:
1347:
1344:
1338:
1330:
1326:
1302:
1294:
1290:
1281:
1262:
1254:
1250:
1229:
1226:
1215:
1212:
1204:
1200:
1174:
1170:
1166:
1163:
1155:
1151:
1136:
1133:
1125:
1121:
1114:
1106:
1090:
1082:
1077:
1073:
1064:
1060:
1051:
1038:
1031:
1026:
1012:
1009:
1003:
995:
991:
967:
959:
955:
951:
948:
940:
936:
927:
923:
900:
896:
887:
883:
879:
860:
852:
848:
824:
820:
813:
805:
801:
797:
792:
784:
776:
772:
767:
761:
758:
753:
745:
733:
730:
724:
717:
716:
715:
713:
694:
691:
688:
685:
682:
679:
676:
673:
665:
653:
650:
642:
638:
632:
624:
621:
613:
609:
605:
597:
585:
582:
576:
569:
568:
567:
565:
561:
542:
539:
531:
527:
523:
520:
512:
508:
504:
496:
484:
481:
473:
469:
461:
460:
459:
457:
456:Serre duality
436:
432:
428:
425:
417:
413:
404:
383:
379:
375:
372:
364:
360:
351:
347:
329:
324:
316:
311:
307:
298:
294:
293:Betti numbers
284:
282:
278:
274:
269:
267:
263:
260:surface as a
252:
248:
246:
245:string theory
242:
238:
234:
230:
226:
222:
218:
214:
210:
206:
201:
199:
180:
177:
172:
168:
164:
159:
155:
151:
146:
142:
138:
133:
129:
121:
120:
119:
118:
114:
110:
106:
102:
99:
95:
91:
87:
83:
79:
75:
69:
66:
59:
58:
56:
52:
48:
44:
40:
26:
21:
7586:Arkani-Hamed
7484:Supergravity
7451:Moduli space
7407:
7348:
7343:Dirac string
7269:Gauge theory
7249:Loop algebra
7186:Black string
7049:GS formalism
6815:
6786:
6760:
6756:
6716:
6702:
6699:"K3 surface"
6666:
6662:
6639:
6594:
6590:
6556:
6512:math/0607339
6502:
6496:
6493:Hulek, Klaus
6476:
6472:
6449:
6445:
6412:
6408:
6368:
6364:
6339:
6335:Demazure, M.
6306:
6264:
6261:Hulek, Klaus
6231:
6210:
6201:
6192:
6183:
6174:
6165:
6156:
6147:
6138:
6129:
6120:
6111:
6102:
6093:
6084:
6075:
6066:
6057:
6048:
6039:
6029:
6021:
6015:
6006:
5997:
5988:
5979:
5970:
5961:
5952:
5943:
5934:
5882:
5877:
5814:
5810:
5795:Ernst Kummer
5763:
5734:
5720:
5716:
5712:
5708:
5704:
5700:
5696:
5692:
5684:
5660:
5656:
5652:
5649:
5641:
5637:
5634:
5624:
5594:
5486:
5482:
5474:
5470:
5468:
5459:
5450:
5376:
5335:
5331:
5329:
5324:
5320:
5174:
5164:
5160:
5127:
5123:
5119:
5028:
5027:line bundle
5016:
5012:
5010:
4912:
4848:
4758:general type
4617:
4611:
4605:subset of a
4603:Zariski open
4572:
4536:
4502:
4498:
4465:
4461:
4460:+ 1, and so
4457:
4453:
4451:
4446:
4442:
4438:
4434:
4430:
4422:
4410:
4408:
4403:
4399:
4337:
4333:
4326:
4323:
4319:
4315:
4313:
4087:
4083:
4079:
4077:
4009:
4005:
4001:
3904:
3876:
3755:
3750:
3681:
3673:
3672:. The space
3554:
3550:
3548:
3509:
3505:
3501:
3497:
3491:
3482:
3470:
3462:
3460:
3450:
3411:
3378:
3376:
3296:
3294:
3187:
3081:
3028:
2912:
2842:
2764:
2760:
2758:
2749:
2741:
2697:
2634:
2611:
2606:is always a
2603:
2601:
2588:
2584:
2580:
2576:
2572:
2569:Picard group
2566:
2480:
2476:
2395:
2387:
2354:
2271:
2267:
2259:
2257:double cover
2188:homeomorphic
2179:
2130:
2025:
1954:
1941:
1767:computed on
1648:Yum-Tong Siu
1451:
1027:
877:
839:
709:
563:
557:
349:
345:
290:
270:
258:
249:
221:general type
205:complex tori
202:
195:
90:irregularity
77:
71:
61:
36:
32:
7946:Silverstein
7446:Orientifold
7181:Black holes
7176:Black brane
7133:Dual photon
6857:K3 database
6812:Weil, André
6763:: 249â260,
6452:: 171â232,
6411:, SĂ©rie 4,
6371:(1): 7â20,
5885:Weil (1958)
5319:. Call the
5177:convex cone
4723:Klaus Hulek
4661:unirational
4616:. For each
4318:K3 surface
2239:John Morgan
2178:4-manifold
1959:cup product
882:Chern class
233:4-manifolds
74:mathematics
25:real points
8071:Categories
7966:Strominger
7961:Steinhardt
7956:Staudacher
7871:Polchinski
7821:Nanopoulos
7781:Mandelstam
7761:Kontsevich
7601:Berenstein
7529:Holography
7509:Superspace
7408:K3 surface
7367:Worldsheet
7282:Instantons
6910:Background
6787:Astérisque
6769:40.0683.03
6485:40.0685.01
6458:25.1212.02
6221:References
5650:Weyl group
5464:reflection
5128:(â2)-curve
5031:such that
4336:such that
2276:hyperplane
2162:E8 lattice
2026:K3 lattice
1660:Ricci-flat
1626:Properties
1028:Next, the
273:projective
255:Definition
78:K3 surface
65:Weil (1958
8001:Veneziano
7881:Rajaraman
7776:Maldacena
7666:Gopakumar
7616:Dijkgraaf
7611:Curtright
7277:Anomalies
7156:NS5-brane
7077:U-duality
7072:S-duality
7067:T-duality
6709:EMS Press
6604:1308.5667
6479:: 25â28,
5857:≥
5671:Δ
5605:ρ
5572:⋅
5549:Δ
5546:∈
5523:Δ
5500:≥
5497:ρ
5431:−
5403:
5397:∈
5385:Δ
5359:−
5261:−
5258:ρ
5224:⊗
5212:
5074:−
4771:≥
4674:≤
4586:≥
4575:for each
4383:−
4316:polarized
4285:Ω
4218:⊂
4206:Ω
4060:Λ
4043:→
4037:Λ
3983:⊗
3980:Λ
3977:≅
3947:⊂
3935:Ω
3888:→
3848:¯
3840:⋅
3803:⊗
3800:Λ
3791:∈
3730:Λ
3701:Λ
3678:Hausdorff
3655:⊕
3647:⊕
3639:⊕
3628:−
3609:Λ
3549:Define a
3508:, unless
3425:→
3316:
3310:∈
3238:χ
3202:→
3168:ρ
3165:−
3139:≤
3136:ρ
3110:−
3107:ρ
3063:ρ
3060:−
3037:ρ
2972:
2966:∈
2937:−
2934:ρ
2889:
2883:∈
2728:ρ
2708:ρ
2672:≅
2621:ρ
2599:theorem.
2421:±
2371:−
2368:↦
2212:×
2184:signature
2112:⊕
2104:⊕
2096:⊕
2085:−
1996:≅
1913:≅
1901:Ω
1856:≅
1839:Ω
1402:−
1390:∑
1374:χ
1192:, and so
1148:→
1118:→
1103:gives an
1088:→
1083:∗
1070:→
1057:→
1042:→
952:−
725:χ
680:−
622:−
610:∑
577:χ
321:Ω
8056:Zwiebach
8011:Verlinde
8006:Verlinde
7981:Townsend
7976:Susskind
7911:Sagnotti
7876:Polyakov
7831:Nekrasov
7796:Minwalla
7791:Martinec
7756:Knizhnik
7751:Klebanov
7746:Kapustin
7711:'t Hooft
7646:Fischler
7581:AganagiÄ
7552:M-theory
7441:Conifold
7436:Orbifold
7419:manifold
7360:Geometry
7166:M5-brane
7161:M2-brane
7098:Graviton
7014:F-theory
6859:for the
6778:(2008),
6748:(1909),
6629:28495199
6554:(2016),
6545:14877568
6467:(1909),
6440:(1893),
6403:(1975),
6393:24693572
6337:(1985),
6269:Springer
5900:See also
5896:(1975).
5625:isolated
3467:uniruled
2400:Jacobian
2250:Examples
2176:oriented
2129:, where
1873:, hence
888:. Since
96:means a
7986:Trivedi
7971:Sundrum
7936:Shenker
7926:Seiberg
7921:Schwarz
7891:Randall
7851:Novikov
7841:Nielsen
7826:NÄstase
7736:Kallosh
7721:Gibbons
7661:Gliozzi
7651:Friedan
7641:Ferrara
7626:Douglas
7621:Distler
7171:S-brane
7151:D-brane
7108:Tachyon
7103:Dilaton
6917:Strings
6838:0537935
6805:2487743
6739:2136212
6691:0284440
6671:Bibcode
6648:2310254
6621:3263959
6579:3586372
6537:2336040
6517:Bibcode
6431:0447635
6385:2312974
6353:0785216
6320:0728990
6295:2030225
6250:1479699
5760:History
5683:. Then
5473:in Pic(
5338:) with
5334:of Pic(
4425:in the
3684:by the
3551:marking
3465:is not
2488:quadric
2398:is the
2262:of the
2160:is the
1759:on the
884:of the
876:is the
401:of the
55:Kashmir
47:Kodaira
8051:Zumino
8046:Zaslow
8031:Yoneya
8021:Witten
7941:Siegel
7916:Scherk
7886:Ramond
7861:Ooguri
7786:Marolf
7741:Kaluza
7726:Kachru
7716:HoĆava
7706:Harvey
7701:Hanson
7686:Gubser
7676:Greene
7606:Bousso
7591:Atiyah
7143:Branes
6953:Theory
6836:
6826:
6803:
6793:
6767:
6737:
6727:
6689:
6646:
6627:
6619:
6577:
6567:
6543:
6535:
6483:
6456:
6429:
6391:
6383:
6351:
6318:
6293:
6283:
6248:
5883:André
5872:, and
5817:â2 in
5644:up to
4756:is of
4614:(2,19)
4505:â2 in
4400:degree
2390:. The
1761:moduli
1452:Since
840:where
405:group
266:2-form
225:curves
101:proper
98:smooth
63:André
43:KĂ€hler
39:Kummer
7991:Turok
7901:RoÄek
7866:Ovrut
7856:Olive
7836:Neveu
7816:Myers
7811:Mukhi
7801:Moore
7771:Linde
7766:Klein
7691:Gukov
7681:Gross
7671:Green
7656:Gates
7636:Dvali
7596:Banks
6783:(PDF)
6753:(PDF)
6625:S2CID
6599:arXiv
6561:(PDF)
6541:S2CID
6507:arXiv
6448:, 2,
6389:S2CID
6236:arXiv
5927:Notes
5707:(Pic(
5695:(Pic(
5687:is a
5659:(Pic(
5561:with
5451:roots
4946:meet
4651:is a
4324:genus
4090:from
3337:with
3128:with
2993:with
2470:nodes
2309:genus
2190:to a
2020:is a
1961:) on
1646:, by
562:) of
279:(the
94:field
8016:Wess
7996:Vafa
7906:Rohm
7806:Motl
7731:Kaku
7696:Guth
7631:Duff
6824:ISBN
6791:ISBN
6725:ISBN
6565:ISBN
6281:ISBN
5578:>
5458:Pic(
5023:and
4406:â2.
4082:and
3853:>
3756:The
3477:and
3086:and
3011:>
2913:The
2843:even
2759:The
2597:GAGA
2571:Pic(
2567:The
2255:The
1665:The
1516:and
880:-th
566:is:
291:The
243:and
88:and
8026:Yau
7951:SÆĄn
7931:Sen
6765:JFM
6679:doi
6609:doi
6525:doi
6503:169
6481:JFM
6454:JFM
6417:doi
6373:doi
6273:doi
5756:).
5711:))/
5691:of
5481:of
5400:Pic
5209:Pic
5122:of
5025:big
5021:nef
4786:or
4760:if
4689:or
4663:if
4413:is
4322:of
4257:to
4134:to
3485:is
3381:in
3313:Pic
2969:Pic
2886:Pic
2595:'s
2311:2.)
2278:in
2170:'s
227:or
196:in
72:In
53:in
8073::
7327:,
7320:,
7313:,
7306:,
6834:MR
6832:,
6801:MR
6799:,
6785:,
6761:68
6759:,
6755:,
6735:MR
6733:,
6723:,
6719:,
6707:,
6701:,
6687:MR
6685:,
6677:,
6665:,
6657:;
6644:MR
6623:,
6617:MR
6615:,
6607:,
6595:90
6593:,
6575:MR
6573:,
6539:,
6533:MR
6531:,
6523:,
6515:,
6501:,
6477:13
6475:,
6471:,
6450:44
6444:,
6427:MR
6425:,
6407:,
6387:,
6381:MR
6379:,
6369:16
6367:,
6363:,
6349:MR
6347:,
6333:;
6329:;
6316:MR
6291:MR
6289:,
6279:,
6271:,
6259:;
6246:MR
6244:,
6026:;
5880:.
5801:,
5797:,
5747:ĂE
5727:.
5327:.
5206::=
4846:.
4830:61
4824:59
4818:58
4812:55
4806:51
4800:47
4774:63
4709:20
4703:18
4677:13
4620:,
4612:SO
4534:.
4449:.
4314:A
4306:.
3253:24
3180:.
3162:20
3142:11
3057:20
3026:.
2910:.
2682:22
2346:A
2050:19
2040:II
2006:22
1923:20
1710:20
1622:.
1610:22
1348:24
1282:,
1025:.
1013:24
762:12
695:2.
606::=
543:1.
458:,
247:.
239:,
200:.
51:K2
45:,
41:,
7417:2
7415:G
7384:?
7349:p
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7331:8
7329:E
7324:7
7322:E
7317:6
7315:E
7310:4
7308:F
7303:2
7301:G
7299:(
6895:e
6888:t
6881:v
6681::
6673::
6667:5
6611::
6601::
6527::
6519::
6509::
6419::
6413:8
6375::
6275::
6238::
6034:.
5922:.
5878:g
5860:3
5854:g
5832:g
5827:P
5815:g
5811:g
5779:3
5774:P
5752:(
5749:8
5745:8
5721:X
5717:X
5713:W
5709:X
5705:O
5701:X
5697:X
5693:O
5685:W
5661:X
5657:O
5653:W
5642:X
5638:X
5611:2
5608:=
5595:X
5581:0
5575:u
5569:A
5543:u
5503:3
5487:X
5483:X
5475:X
5471:A
5460:X
5437:}
5434:2
5428:=
5423:2
5419:u
5415::
5412:)
5409:X
5406:(
5394:u
5391:{
5388:=
5362:2
5356:=
5351:2
5347:u
5336:X
5332:u
5325:X
5303:)
5300:X
5297:(
5292:1
5288:N
5267:)
5264:1
5255:,
5252:1
5249:(
5228:R
5221:)
5218:X
5215:(
5203:)
5200:X
5197:(
5192:1
5188:N
5165:Y
5161:g
5145:1
5140:P
5124:X
5120:Y
5104:g
5099:P
5077:2
5071:g
5068:2
5065:=
5060:2
5056:)
5052:L
5049:(
5044:1
5040:c
5029:L
5017:g
4994:g
4988:F
4963:g
4957:F
4932:h
4926:F
4913:g
4897:g
4891:F
4866:g
4860:F
4827:,
4821:,
4815:,
4809:,
4803:,
4797:=
4794:g
4768:g
4742:g
4736:F
4706:,
4700:=
4697:g
4671:g
4637:g
4631:F
4618:g
4589:2
4583:g
4573:g
4557:g
4551:F
4520:g
4515:P
4503:g
4499:X
4483:g
4478:P
4466:X
4462:L
4458:g
4454:L
4447:g
4443:L
4441:,
4439:X
4435:g
4431:L
4429:|
4423:C
4411:L
4404:g
4402:2
4386:2
4380:g
4377:2
4374:=
4369:2
4365:)
4361:L
4358:(
4353:1
4349:c
4338:L
4334:L
4327:g
4320:X
4294:)
4289:2
4281:,
4278:Y
4275:(
4270:0
4266:H
4245:)
4241:C
4237:,
4234:X
4231:(
4226:2
4222:H
4215:)
4210:2
4202:,
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4191:0
4187:H
4166:)
4162:Z
4158:,
4155:Y
4152:(
4147:2
4143:H
4122:)
4118:Z
4114:,
4111:X
4108:(
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4099:H
4084:Y
4080:X
4063:)
4057:(
4054:O
4050:/
4046:D
4040:)
4034:(
4031:O
4027:/
4023:N
4006:N
4002:D
3987:C
3974:)
3970:C
3966:,
3963:X
3960:(
3955:2
3951:H
3944:)
3939:2
3931:,
3928:X
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3905:X
3891:D
3885:N
3862:.
3859:}
3856:0
3845:u
3837:u
3833:,
3830:0
3827:=
3822:2
3818:u
3814::
3811:)
3807:C
3797:(
3794:P
3788:u
3785:{
3782:=
3779:D
3751:N
3733:)
3727:(
3724:O
3704:)
3698:(
3695:O
3682:N
3674:N
3658:3
3651:U
3642:2
3635:)
3631:1
3625:(
3620:8
3616:E
3612:=
3589:)
3585:Z
3581:,
3578:X
3575:(
3570:2
3566:H
3555:X
3527:2
3522:C
3510:X
3506:X
3502:X
3498:X
3483:X
3471:X
3463:X
3451:L
3435:1
3430:P
3422:X
3412:L
3396:3
3391:P
3379:X
3358:0
3355:=
3350:2
3346:u
3325:)
3322:X
3319:(
3307:u
3297:X
3278:1
3274:I
3250:=
3247:)
3244:X
3241:(
3212:1
3207:P
3199:X
3116:)
3113:1
3104:,
3101:1
3098:(
3014:0
3006:2
3002:u
2981:)
2978:X
2975:(
2963:u
2943:)
2940:1
2931:,
2928:1
2925:(
2898:)
2895:X
2892:(
2880:u
2858:2
2854:u
2821:)
2817:Z
2813:,
2810:X
2807:(
2802:2
2798:H
2776:C
2765:X
2750:p
2742:X
2698:X
2677:Z
2669:)
2665:Z
2661:,
2658:X
2655:(
2650:2
2646:H
2635:X
2604:X
2589:X
2585:X
2581:X
2577:X
2573:X
2558:.
2537:5
2532:P
2505:4
2500:P
2481:Y
2477:Y
2472:.
2454:3
2449:P
2427:)
2424:1
2418:(
2414:/
2410:A
2396:A
2388:A
2374:a
2365:a
2355:A
2329:3
2324:P
2293:2
2288:P
2272:X
2268:g
2260:X
2234:.
2220:2
2216:S
2207:2
2203:S
2180:X
2164:.
2146:8
2142:E
2131:U
2115:3
2108:U
2099:2
2092:)
2088:1
2082:(
2077:8
2073:E
2047:,
2044:3
2001:Z
1993:)
1989:Z
1985:,
1982:X
1979:(
1974:2
1970:H
1955:X
1948:.
1942:p
1918:C
1910:)
1905:X
1897:,
1894:X
1891:(
1886:1
1882:H
1860:C
1853:)
1848:2
1843:X
1835:;
1832:X
1829:(
1824:0
1820:H
1799:)
1795:Z
1791:;
1788:X
1785:(
1780:2
1776:H
1738:1
1727:0
1722:0
1715:1
1705:1
1698:0
1693:0
1682:1
1639:.
1607:=
1604:)
1601:X
1598:(
1593:2
1589:b
1568:0
1565:=
1562:)
1559:X
1556:(
1551:3
1547:b
1543:=
1540:)
1537:X
1534:(
1529:1
1525:b
1504:1
1501:=
1498:)
1495:X
1492:(
1487:4
1483:b
1479:=
1476:)
1473:X
1470:(
1465:0
1461:b
1437:.
1434:)
1431:X
1428:(
1423:i
1419:b
1413:i
1409:)
1405:1
1399:(
1394:i
1386:=
1383:)
1380:X
1377:(
1345:=
1342:)
1339:X
1336:(
1331:2
1327:c
1306:)
1303:X
1300:(
1295:3
1291:b
1266:)
1263:X
1260:(
1255:1
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1230:0
1227:=
1224:)
1220:Z
1216:,
1213:X
1210:(
1205:1
1201:H
1180:)
1175:X
1171:O
1167:,
1164:X
1161:(
1156:1
1152:H
1145:)
1141:Z
1137:,
1134:X
1131:(
1126:1
1122:H
1115:0
1091:0
1078:X
1074:O
1065:X
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1052:X
1047:Z
1039:0
1010:=
1007:)
1004:X
1001:(
996:2
992:c
971:)
968:X
965:(
960:1
956:c
949:=
946:)
941:X
937:K
933:(
928:1
924:c
901:X
897:K
878:i
864:)
861:X
858:(
853:i
849:c
825:,
821:)
817:)
814:X
811:(
806:2
802:c
798:+
793:2
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785:X
782:(
777:1
773:c
768:(
759:1
754:=
751:)
746:X
740:O
734:,
731:X
728:(
692:=
689:1
686:+
683:0
677:1
674:=
671:)
666:X
660:O
654:,
651:X
648:(
643:i
639:h
633:i
629:)
625:1
619:(
614:i
603:)
598:X
592:O
586:,
583:X
580:(
564:X
540:=
537:)
532:X
528:K
524:,
521:X
518:(
513:0
509:h
505:=
502:)
497:X
491:O
485:,
482:X
479:(
474:2
470:h
442:)
437:X
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429:,
426:X
423:(
418:1
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384:X
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373:X
370:(
365:1
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350:X
348:(
346:q
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317:=
312:X
308:K
181:0
178:=
173:4
169:w
165:+
160:4
156:z
152:+
147:4
143:y
139:+
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57:.
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