1151:
527:(as an algebraic vector bundle with flat connection), because its solutions do not have moderate growth at ∞. This shows the need to restrict to flat connections with regular singularities in the Riemann–Hilbert correspondence. On the other hand, if we work with holomorphic (rather than algebraic) vector bundles with flat connection on a noncompact complex manifold such as
64:, the Riemann-Hilbert correspondence provides a complex analytic isomorphism between two of the three natural algebraic structures on the moduli spaces, and so is naturally viewed as a nonabelian analogue of the comparison isomorphism between De Rham cohomology and singular/Betti cohomology.
214:
has dimension one (a complex algebraic curve) then there is a more general
Riemann–Hilbert correspondence for algebraic connections with no regularity assumption (or for holonomic D-modules with no regularity assumption) described in Malgrange (1991), the
596:-sheaves and left (resp. right) modules with a Frobenius (resp. Cartier) action. This can be regarded as the positive characteristic analogue of the classical theory, where one can find a similar interplay of constructive vs. perverse t-structures.
535:, then the notion of regular singularities is not defined. A much more elementary theorem than the Riemann–Hilbert correspondence states that flat connections on holomorphic vector bundles are determined up to isomorphism by their monodromy.
20:
refers to the correspondence between regular singular flat connections on algebraic vector bundles and representations of the fundamental group, and more generally to one of several generalizations of this. The original setting appearing in
207:, and a local system on a subvariety is something like a description of possible monodromies, so this correspondence can be thought of as describing certain systems of differential equations in terms of the monodromies of their solutions.
499:
281:
395:
329:− {0}. That means that the equation has nontrivial monodromy. Explicitly, the monodromy of this equation is the 1-dimensional representation of the fundamental group
113:
The condition of regular singularities means that locally constant sections of the bundle (with respect to the flat connection) have moderate growth at points of
523:, the monodromy of this flat connection is trivial. But this flat connection is not isomorphic to the obvious flat connection on the trivial line bundle over
1111:
110:. Thus such connections give a purely algebraic way to access the finite dimensional representations of the topological fundamental group.
424:, the equation does not have regular singularities at ∞. (This can also be seen by rewriting the equation in terms of the variable
931:
764:(1980), "Faisceaux constructibles et systèmes holonômes d'équations aux dérivées partielles linéaires à points singuliers réguliers",
966:
784:
971:
86:
called the local solutions functor, that is an equivalence from the category of flat connections on algebraic vector bundles on
703:
438:
168:
By considering the irreducible elements of each category, this gives a 1:1 correspondence between isomorphism classes of
52:(1970, generalizing existing work in the case of Riemann surfaces) and more generally for regular holonomic D-modules by
1076:
1061:
981:
683:(Gives explicit representation of Riemann–Hilbert correspondence for Milnor fiber of isolated hypersurface singularity)
233:
1121:
866:
833:
22:
1136:
1177:
420:. Since these solutions do not have polynomial growth on some sectors around the point ∞ in the projective line
1182:
1131:
1126:
1101:
956:
924:
736:
102:
connected, the category of local systems is also equivalent to the category of complex representations of the
45:(linear and having very special properties for their solutions) and possible monodromies of their solutions.
42:
26:
1066:
951:
1091:
605:
357:
1116:
1096:
504:
The pole of order 2 in the coefficients means that the equation does not have regular singularities at
1187:
1155:
917:
547:
631:
Bhatt, Bhargav; Lurie, Jacob (2019), "A Riemann-Hilbert correspondence in positive characteristic",
1106:
996:
961:
588:
More generally, there are equivalences of categories between constructible (resp. perverse) étale
1016:
184:
61:
618:
Emerton, Matthew; Kisin, Mark (2004), "The
Riemann-Hilbert correspondence for unit F-crystals",
351:
To see the need for the hypothesis of regular singularities, consider the differential equation
1031:
887:
578:
509:
302:
1056:
154:
91:
408:). This equation corresponds to a flat connection on the trivial algebraic line bundle over
903:
876:
843:
810:
773:
746:
713:
662:
8:
1081:
991:
976:
855:
Complex analysis, microlocal calculus and relativistic quantum theory (Les
Houches, 1979)
582:
562:
1001:
666:
640:
143:
1071:
1011:
862:
829:
817:
750:
732:
699:
670:
103:
1006:
986:
940:
883:
850:
796:
780:
761:
650:
158:
57:
53:
38:
25:
was for the
Riemann sphere, where it was about the existence of systems of linear
1051:
1026:
899:
872:
858:
839:
806:
769:
742:
728:
709:
658:
48:
Such a result was proved for algebraic connections with regular singularities by
34:
94:
to the category of local systems of finite-dimensional complex vector spaces on
1041:
1036:
801:
720:
695:
291:
49:
654:
566:
1171:
561:) establish a Riemann-Hilbert correspondence that asserts in particular that
825:
344:
in which the generator (a loop around the origin) acts by multiplication by
1021:
687:
192:
41:
of dimension > 1. There is a correspondence between certain systems of
33:
representations. First the
Riemann sphere may be replaced by an arbitrary
754:
1046:
142:
called the de Rham functor, that is an equivalence from the category of
30:
909:
645:
200:
146:
227:
An example where the theorem applies is the differential equation
37:
and then, in higher dimensions, Riemann surfaces are replaced by
789:
Publications of the
Research Institute for Mathematical Sciences
557:(later developed further under less restrictive assumptions in
129:
is compact, the condition of regular singularities is vacuous.
577:-coefficients can be computed in terms of the action of the
694:, Perspectives in Mathematics, vol. 2, Boston, MA:
203:
is something like a system of differential equations on
82:(for regular singular connections): there is a functor
725:Équations différentielles à points singuliers réguliers
494:{\displaystyle {\frac {df}{dw}}=-{\frac {1}{w^{2}}}f.}
309:. The local solutions of the equation are of the form
138:(for regular holonomic D-modules): there is a functor
441:
360:
236:
822:Équations différentielles à coefficients polynomiaux
785:"The Riemann-Hilbert problem for holonomic systems"
493:
389:
275:
1169:
766:Séminaire Goulaouic-Schwartz, 1979–80, Exposé 19
412:. The solutions of the equation are of the form
276:{\displaystyle {\frac {df}{dz}}={\frac {a}{z}}f}
187:complexes of irreducible closed subvarieties of
727:, Lecture Notes in Mathematics, vol. 163,
853:(1980), "Sur le problėme de Hilbert-Riemann",
60:(1980, 1984) independently. In the setting of
925:
301:is a fixed complex number. This equation has
967:Grothendieck–Hirzebruch–Riemann–Roch theorem
617:
554:
857:, Lecture Notes in Physics, vol. 126,
932:
918:
630:
558:
290: − {0} (that is, on the nonzero
1112:Riemann–Roch theorem for smooth manifolds
824:, Progress in Mathematics, vol. 96,
816:
800:
779:
760:
644:
882:
849:
719:
217:Riemann–Hilbert–Birkhoff correspondence
76:is a smooth complex algebraic variety.
1170:
538:
325:cannot be made well-defined on all of
913:
888:"Une autre équivalence de catégories"
686:
519:are defined on the whole affine line
321:is not an integer, then the function
939:
121:is an algebraic compactification of
172:irreducible holonomic D-modules on
13:
1077:Riemannian connection on a surface
982:Measurable Riemann mapping theorem
768:, Palaiseau: École Polytechnique,
390:{\displaystyle {\frac {df}{dz}}=f}
305:at 0 and ∞ in the projective line
14:
1199:
676:
404:(that is, on the complex numbers
191:with coefficients in irreducible
1150:
1149:
633:Cambridge Journal of Mathematics
1062:Riemann's differential equation
972:Hirzebruch–Riemann–Roch theorem
1087:Riemann–Hilbert correspondence
957:Generalized Riemann hypothesis
136:Riemann–Hilbert correspondence
80:Riemann–Hilbert correspondence
43:partial differential equations
27:regular differential equations
23:Hilbert's twenty-first problem
18:Riemann–Hilbert correspondence
1:
1122:Riemann–Siegel theta function
611:
286:on the punctured affine line
1137:Riemann–von Mangoldt formula
132:More generally there is the
67:
7:
599:
222:
176:with regular singularities,
10:
1204:
1132:Riemann–Stieltjes integral
1127:Riemann–Silberstein vector
1102:Riemann–Liouville integral
555:Emerton & Kisin (2004)
1145:
1067:Riemann's minimal surface
947:
655:10.4310/CJM.2019.v7.n1.a3
340: − {0}) =
16:In mathematics, the term
1092:Riemann–Hilbert problems
997:Riemann curvature tensor
962:Grand Riemann hypothesis
952:Cauchy–Riemann equations
802:10.2977/prims/1195181610
559:Bhatt & Lurie (2019)
1017:Riemann mapping theorem
606:Riemann–Hilbert problem
185:intersection cohomology
62:nonabelian Hodge theory
1178:Differential equations
1117:Riemann–Siegel formula
1097:Riemann–Lebesgue lemma
1032:Riemann series theorem
892:Compositio Mathematica
579:Frobenius endomorphism
495:
391:
277:
125:. In particular, when
1183:Representation theory
1057:Riemann zeta function
496:
392:
303:regular singularities
278:
155:regular singularities
92:regular singularities
1107:Riemann–Roch theorem
515:Since the functions
439:
358:
234:
1082:Riemannian geometry
992:Riemann Xi function
977:Local zeta function
861:, pp. 90–110,
692:Algebraic D-Modules
679:Sheaves in Topology
583:coherent cohomology
432:, where it becomes
400:on the affine line
157:to the category of
1002:Riemann hypothesis
818:Malgrange, Bernard
681:, pp. 206–207
677:Dimca, Alexandru,
539:In characteristic
508:= 0, according to
491:
387:
273:
1165:
1164:
1072:Riemannian circle
1012:Riemann invariant
884:Mebkhout, Zoghman
851:Mebkhout, Zoghman
781:Kashiwara, Masaki
762:Kashiwara, Masaki
705:978-0-12-117740-9
483:
460:
379:
268:
255:
104:fundamental group
56:(1980, 1984) and
39:complex manifolds
1195:
1188:Bernhard Riemann
1153:
1152:
1007:Riemann integral
987:Riemann (crater)
941:Bernhard Riemann
934:
927:
920:
911:
910:
906:
879:
846:
813:
804:
776:
757:
716:
682:
673:
648:
627:
563:étale cohomology
500:
498:
497:
492:
484:
482:
481:
469:
461:
459:
451:
443:
428: := 1/
396:
394:
393:
388:
380:
378:
370:
362:
332:
282:
280:
279:
274:
269:
261:
256:
254:
246:
238:
159:perverse sheaves
58:Zoghman Mebkhout
54:Masaki Kashiwara
29:with prescribed
1203:
1202:
1198:
1197:
1196:
1194:
1193:
1192:
1168:
1167:
1166:
1161:
1141:
1052:Riemann surface
1027:Riemann problem
943:
938:
869:
859:Springer-Verlag
836:
739:
729:Springer-Verlag
721:Deligne, Pierre
706:
639:(1–2): 71–217,
614:
602:
546:For schemes in
544:
510:Fuchs's theorem
477:
473:
468:
452:
444:
442:
440:
437:
436:
371:
363:
361:
359:
356:
355:
335:
330:
292:complex numbers
260:
247:
239:
237:
235:
232:
231:
225:
70:
35:Riemann surface
12:
11:
5:
1201:
1191:
1190:
1185:
1180:
1163:
1162:
1160:
1159:
1146:
1143:
1142:
1140:
1139:
1134:
1129:
1124:
1119:
1114:
1109:
1104:
1099:
1094:
1089:
1084:
1079:
1074:
1069:
1064:
1059:
1054:
1049:
1044:
1042:Riemann sphere
1039:
1037:Riemann solver
1034:
1029:
1024:
1019:
1014:
1009:
1004:
999:
994:
989:
984:
979:
974:
969:
964:
959:
954:
948:
945:
944:
937:
936:
929:
922:
914:
908:
907:
880:
867:
847:
834:
814:
795:(2): 319–365,
777:
758:
737:
717:
704:
696:Academic Press
684:
674:
628:
613:
610:
609:
608:
601:
598:
548:characteristic
543:
537:
502:
501:
490:
487:
480:
476:
472:
467:
464:
458:
455:
450:
447:
416:for constants
398:
397:
386:
383:
377:
374:
369:
366:
333:
313:for constants
284:
283:
272:
267:
264:
259:
253:
250:
245:
242:
224:
221:
197:
196:
178:
177:
69:
66:
50:Pierre Deligne
9:
6:
4:
3:
2:
1200:
1189:
1186:
1184:
1181:
1179:
1176:
1175:
1173:
1158:
1157:
1148:
1147:
1144:
1138:
1135:
1133:
1130:
1128:
1125:
1123:
1120:
1118:
1115:
1113:
1110:
1108:
1105:
1103:
1100:
1098:
1095:
1093:
1090:
1088:
1085:
1083:
1080:
1078:
1075:
1073:
1070:
1068:
1065:
1063:
1060:
1058:
1055:
1053:
1050:
1048:
1045:
1043:
1040:
1038:
1035:
1033:
1030:
1028:
1025:
1023:
1020:
1018:
1015:
1013:
1010:
1008:
1005:
1003:
1000:
998:
995:
993:
990:
988:
985:
983:
980:
978:
975:
973:
970:
968:
965:
963:
960:
958:
955:
953:
950:
949:
946:
942:
935:
930:
928:
923:
921:
916:
915:
912:
905:
901:
897:
893:
889:
885:
881:
878:
874:
870:
868:3-540-09996-4
864:
860:
856:
852:
848:
845:
841:
837:
835:0-8176-3556-4
831:
827:
823:
819:
815:
812:
808:
803:
798:
794:
790:
786:
782:
778:
775:
771:
767:
763:
759:
756:
752:
748:
744:
740:
734:
730:
726:
722:
718:
715:
711:
707:
701:
697:
693:
689:
688:Borel, Armand
685:
680:
675:
672:
668:
664:
660:
656:
652:
647:
642:
638:
634:
629:
625:
621:
616:
615:
607:
604:
603:
597:
595:
591:
586:
584:
580:
576:
572:
568:
567:étale sheaves
564:
560:
556:
552:
549:
542:
536:
534:
530:
526:
522:
518:
513:
511:
507:
488:
485:
478:
474:
470:
465:
462:
456:
453:
448:
445:
435:
434:
433:
431:
427:
423:
419:
415:
411:
407:
403:
384:
381:
375:
372:
367:
364:
354:
353:
352:
349:
347:
343:
339:
328:
324:
320:
316:
312:
308:
304:
300:
297:− {0}). Here
296:
293:
289:
270:
265:
262:
257:
251:
248:
243:
240:
230:
229:
228:
220:
218:
213:
208:
206:
202:
194:
193:local systems
190:
186:
183:
182:
181:
175:
171:
170:
169:
166:
164:
160:
156:
152:
148:
145:
141:
137:
133:
130:
128:
124:
120:
116:
111:
109:
105:
101:
97:
93:
89:
85:
81:
77:
75:
72:Suppose that
65:
63:
59:
55:
51:
46:
44:
40:
36:
32:
28:
24:
19:
1154:
1086:
1022:Riemann form
898:(1): 63–88,
895:
891:
854:
821:
792:
788:
765:
724:
691:
678:
636:
632:
623:
619:
593:
589:
587:
574:
570:
550:
545:
540:
532:
528:
524:
520:
516:
514:
505:
503:
429:
425:
421:
417:
413:
409:
405:
401:
399:
350:
345:
341:
337:
326:
322:
318:
314:
310:
306:
298:
294:
287:
285:
226:
216:
211:
210:In the case
209:
204:
198:
188:
179:
173:
167:
162:
150:
139:
135:
134:
131:
126:
122:
118:
114:
112:
107:
99:
95:
87:
83:
79:
78:
73:
71:
47:
17:
15:
1047:Riemann sum
1172:Categories
826:Birkhäuser
738:3540051902
646:1711.04148
620:Astérisque
612:References
671:119147066
466:−
147:D-modules
144:holonomic
68:Statement
31:monodromy
1156:Category
886:(1984),
820:(1991),
783:(1984),
723:(1970),
690:(1987),
600:See also
223:Examples
201:D-module
117:, where
904:0734785
877:0579742
844:1117227
811:0743382
774:0600704
747:0417174
714:0882000
663:3922360
626:: 1–268
553:>0,
902:
875:
865:
842:
832:
809:
772:
755:169357
753:
745:
735:
712:
702:
669:
661:
98:. For
667:S2CID
641:arXiv
569:with
317:. If
153:with
115:Y − X
90:with
863:ISBN
830:ISBN
751:OCLC
733:ISBN
700:ISBN
180:and
797:doi
651:doi
624:293
581:on
565:of
512:.)
161:on
149:on
106:of
84:Sol
1174::
900:MR
896:51
894:,
890:,
873:MR
871:,
840:MR
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828:,
807:MR
805:,
793:20
791:,
787:,
770:MR
749:,
743:MR
741:,
731:,
710:MR
708:,
698:,
665:,
659:MR
657:,
649:,
635:,
622:,
585:.
531:=
517:ce
414:ce
348:.
311:cz
219:.
199:A
165:.
140:DR
933:e
926:t
919:v
799::
653::
643::
637:7
594:p
592:/
590:Z
575:p
573:/
571:Z
551:p
541:p
533:C
529:A
525:A
521:A
506:w
489:.
486:f
479:2
475:w
471:1
463:=
457:w
454:d
449:f
446:d
430:z
426:w
422:P
418:c
410:A
406:C
402:A
385:f
382:=
376:z
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368:f
365:d
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342:Z
338:A
336:(
334:1
331:π
327:C
323:z
319:a
315:c
307:P
299:a
295:C
288:A
271:f
266:z
263:a
258:=
252:z
249:d
244:f
241:d
212:X
205:X
195:.
189:X
174:X
163:X
151:X
127:X
123:X
119:Y
108:X
100:X
96:X
88:X
74:X
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