1162:
538:(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
75:, 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.
225:
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
607:-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.
546:, 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.
31:
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
218:, 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.
510:
292:
406:
340:− {0}. That means that the equation has nontrivial monodromy. Explicitly, the monodromy of this equation is the 1-dimensional representation of the fundamental group
124:
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
534:, 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
17:
1122:
121:. Thus such connections give a purely algebraic way to access the finite dimensional representations of the topological fundamental group.
435:, the equation does not have regular singularities at ∞. (This can also be seen by rewriting the equation in terms of the variable
942:
775:(1980), "Faisceaux constructibles et systèmes holonômes d'équations aux dérivées partielles linéaires à points singuliers réguliers",
977:
795:
982:
97:
called the local solutions functor, that is an equivalence from the category of flat connections on algebraic vector bundles on
714:
449:
179:
By considering the irreducible elements of each category, this gives a 1:1 correspondence between isomorphism classes of
63:(1970, generalizing existing work in the case of Riemann surfaces) and more generally for regular holonomic D-modules by
1087:
1072:
992:
694:(Gives explicit representation of Riemann–Hilbert correspondence for Milnor fiber of isolated hypersurface singularity)
244:
1132:
877:
844:
33:
1147:
1188:
431:. Since these solutions do not have polynomial growth on some sectors around the point ∞ in the projective line
1193:
1142:
1137:
1112:
967:
935:
747:
113:
connected, the category of local systems is also equivalent to the category of complex representations of the
56:(linear and having very special properties for their solutions) and possible monodromies of their solutions.
53:
37:
1077:
962:
1102:
616:
368:
1127:
1107:
515:
The pole of order 2 in the coefficients means that the equation does not have regular singularities at
1198:
1166:
928:
558:
642:
Bhatt, Bhargav; Lurie, Jacob (2019), "A Riemann-Hilbert correspondence in positive characteristic",
1117:
1007:
972:
599:
More generally, there are equivalences of categories between constructible (resp. perverse) étale
1027:
195:
72:
629:
Emerton, Matthew; Kisin, Mark (2004), "The
Riemann-Hilbert correspondence for unit F-crystals",
362:
To see the need for the hypothesis of regular singularities, consider the differential equation
1042:
898:
589:
520:
313:
1067:
165:
102:
419:). This equation corresponds to a flat connection on the trivial algebraic line bundle over
914:
887:
854:
821:
784:
757:
724:
673:
8:
1092:
1002:
987:
866:
Complex analysis, microlocal calculus and relativistic quantum theory (Les
Houches, 1979)
593:
573:
1012:
677:
651:
154:
1082:
1022:
873:
840:
828:
761:
743:
710:
681:
114:
1017:
997:
951:
894:
861:
807:
791:
772:
661:
169:
68:
64:
49:
36:
was for the
Riemann sphere, where it was about the existence of systems of linear
1062:
1037:
910:
883:
869:
850:
817:
780:
753:
739:
720:
669:
59:
Such a result was proved for algebraic connections with regular singularities by
45:
105:
to the category of local systems of finite-dimensional complex vector spaces on
1052:
1047:
812:
731:
706:
302:
60:
665:
577:
1182:
572:) establish a Riemann-Hilbert correspondence that asserts in particular that
836:
355:
in which the generator (a loop around the origin) acts by multiplication by
1032:
698:
203:
52:
of dimension > 1. There is a correspondence between certain systems of
44:
representations. First the
Riemann sphere may be replaced by an arbitrary
765:
1057:
153:
called the de Rham functor, that is an equivalence from the category of
41:
920:
656:
211:
157:
238:
An example where the theorem applies is the differential equation
48:
and then, in higher dimensions, Riemann surfaces are replaced by
800:
Publications of the
Research Institute for Mathematical Sciences
568:(later developed further under less restrictive assumptions in
140:
is compact, the condition of regular singularities is vacuous.
588:-coefficients can be computed in terms of the action of the
705:, Perspectives in Mathematics, vol. 2, Boston, MA:
214:
is something like a system of differential equations on
93:(for regular singular connections): there is a functor
736:Équations différentielles à points singuliers réguliers
505:{\displaystyle {\frac {df}{dw}}=-{\frac {1}{w^{2}}}f.}
320:. The local solutions of the equation are of the form
149:(for regular holonomic D-modules): there is a functor
452:
371:
247:
833:Équations différentielles à coefficients polynomiaux
796:"The Riemann-Hilbert problem for holonomic systems"
504:
400:
286:
1180:
777:Séminaire Goulaouic-Schwartz, 1979–80, Exposé 19
423:. The solutions of the equation are of the form
287:{\displaystyle {\frac {df}{dz}}={\frac {a}{z}}f}
198:complexes of irreducible closed subvarieties of
738:, Lecture Notes in Mathematics, vol. 163,
864:(1980), "Sur le problėme de Hilbert-Riemann",
71:(1980, 1984) independently. In the setting of
936:
312:is a fixed complex number. This equation has
978:Grothendieck–Hirzebruch–Riemann–Roch theorem
628:
565:
868:, Lecture Notes in Physics, vol. 126,
943:
929:
641:
569:
301: − {0} (that is, on the nonzero
1123:Riemann–Roch theorem for smooth manifolds
835:, Progress in Mathematics, vol. 96,
827:
811:
790:
771:
655:
893:
860:
730:
228:Riemann–Hilbert–Birkhoff correspondence
87:is a smooth complex algebraic variety.
14:
1181:
549:
336:cannot be made well-defined on all of
924:
899:"Une autre équivalence de catégories"
697:
530:are defined on the whole affine line
332:is not an integer, then the function
950:
132:is an algebraic compactification of
183:irreducible holonomic D-modules on
24:
1088:Riemannian connection on a surface
993:Measurable Riemann mapping theorem
779:, Palaiseau: École Polytechnique,
401:{\displaystyle {\frac {df}{dz}}=f}
316:at 0 and ∞ in the projective line
25:
1210:
687:
415:(that is, on the complex numbers
202:with coefficients in irreducible
1161:
1160:
644:Cambridge Journal of Mathematics
1073:Riemann's differential equation
983:Hirzebruch–Riemann–Roch theorem
1098:Riemann–Hilbert correspondence
968:Generalized Riemann hypothesis
147:Riemann–Hilbert correspondence
91:Riemann–Hilbert correspondence
54:partial differential equations
38:regular differential equations
34:Hilbert's twenty-first problem
29:Riemann–Hilbert correspondence
18:Riemann-Hilbert correspondence
13:
1:
1133:Riemann–Siegel theta function
622:
297:on the punctured affine line
1148:Riemann–von Mangoldt formula
143:More generally there is the
78:
7:
610:
233:
187:with regular singularities,
10:
1215:
1143:Riemann–Stieltjes integral
1138:Riemann–Silberstein vector
1113:Riemann–Liouville integral
566:Emerton & Kisin (2004)
1156:
1078:Riemann's minimal surface
958:
666:10.4310/CJM.2019.v7.n1.a3
351: − {0}) =
27:In mathematics, the term
1103:Riemann–Hilbert problems
1008:Riemann curvature tensor
973:Grand Riemann hypothesis
963:Cauchy–Riemann equations
813:10.2977/prims/1195181610
570:Bhatt & Lurie (2019)
1028:Riemann mapping theorem
617:Riemann–Hilbert problem
196:intersection cohomology
73:nonabelian Hodge theory
1189:Differential equations
1128:Riemann–Siegel formula
1108:Riemann–Lebesgue lemma
1043:Riemann series theorem
903:Compositio Mathematica
590:Frobenius endomorphism
506:
402:
288:
136:. In particular, when
1194:Representation theory
1068:Riemann zeta function
507:
403:
314:regular singularities
289:
166:regular singularities
103:regular singularities
1118:Riemann–Roch theorem
526:Since the functions
450:
369:
245:
1093:Riemannian geometry
1003:Riemann Xi function
988:Local zeta function
872:, pp. 90–110,
703:Algebraic D-Modules
690:Sheaves in Topology
594:coherent cohomology
443:, where it becomes
411:on the affine line
168:to the category of
1013:Riemann hypothesis
829:Malgrange, Bernard
692:, pp. 206–207
688:Dimca, Alexandru,
550:In characteristic
519:= 0, according to
502:
398:
284:
1176:
1175:
1083:Riemannian circle
1023:Riemann invariant
895:Mebkhout, Zoghman
862:Mebkhout, Zoghman
792:Kashiwara, Masaki
773:Kashiwara, Masaki
716:978-0-12-117740-9
494:
471:
390:
279:
266:
115:fundamental group
67:(1980, 1984) and
50:complex manifolds
16:(Redirected from
1206:
1199:Bernhard Riemann
1164:
1163:
1018:Riemann integral
998:Riemann (crater)
952:Bernhard Riemann
945:
938:
931:
922:
921:
917:
890:
857:
824:
815:
787:
768:
727:
693:
684:
659:
638:
574:étale cohomology
511:
509:
508:
503:
495:
493:
492:
480:
472:
470:
462:
454:
439: := 1/
407:
405:
404:
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391:
389:
381:
373:
343:
293:
291:
290:
285:
280:
272:
267:
265:
257:
249:
170:perverse sheaves
69:Zoghman Mebkhout
65:Masaki Kashiwara
40:with prescribed
21:
1214:
1213:
1209:
1208:
1207:
1205:
1204:
1203:
1179:
1178:
1177:
1172:
1152:
1063:Riemann surface
1038:Riemann problem
954:
949:
880:
870:Springer-Verlag
847:
750:
740:Springer-Verlag
732:Deligne, Pierre
717:
650:(1–2): 71–217,
625:
613:
557:For schemes in
555:
521:Fuchs's theorem
488:
484:
479:
463:
455:
453:
451:
448:
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382:
374:
372:
370:
367:
366:
346:
341:
303:complex numbers
271:
258:
250:
248:
246:
243:
242:
236:
81:
46:Riemann surface
23:
22:
15:
12:
11:
5:
1212:
1202:
1201:
1196:
1191:
1174:
1173:
1171:
1170:
1157:
1154:
1153:
1151:
1150:
1145:
1140:
1135:
1130:
1125:
1120:
1115:
1110:
1105:
1100:
1095:
1090:
1085:
1080:
1075:
1070:
1065:
1060:
1055:
1053:Riemann sphere
1050:
1048:Riemann solver
1045:
1040:
1035:
1030:
1025:
1020:
1015:
1010:
1005:
1000:
995:
990:
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975:
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955:
948:
947:
940:
933:
925:
919:
918:
891:
878:
858:
845:
825:
806:(2): 319–365,
788:
769:
748:
728:
715:
707:Academic Press
695:
685:
639:
624:
621:
620:
619:
612:
609:
559:characteristic
554:
548:
513:
512:
501:
498:
491:
487:
483:
478:
475:
469:
466:
461:
458:
427:for constants
409:
408:
397:
394:
388:
385:
380:
377:
344:
324:for constants
295:
294:
283:
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275:
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235:
232:
208:
207:
189:
188:
80:
77:
61:Pierre Deligne
9:
6:
4:
3:
2:
1211:
1200:
1197:
1195:
1192:
1190:
1187:
1186:
1184:
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1159:
1158:
1155:
1149:
1146:
1144:
1141:
1139:
1136:
1134:
1131:
1129:
1126:
1124:
1121:
1119:
1116:
1114:
1111:
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1106:
1104:
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1089:
1086:
1084:
1081:
1079:
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1064:
1061:
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1046:
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1036:
1034:
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1026:
1024:
1021:
1019:
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1011:
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1004:
1001:
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996:
994:
991:
989:
986:
984:
981:
979:
976:
974:
971:
969:
966:
964:
961:
960:
957:
953:
946:
941:
939:
934:
932:
927:
926:
923:
916:
912:
908:
904:
900:
896:
892:
889:
885:
881:
879:3-540-09996-4
875:
871:
867:
863:
859:
856:
852:
848:
846:0-8176-3556-4
842:
838:
834:
830:
826:
823:
819:
814:
809:
805:
801:
797:
793:
789:
786:
782:
778:
774:
770:
767:
763:
759:
755:
751:
745:
741:
737:
733:
729:
726:
722:
718:
712:
708:
704:
700:
699:Borel, Armand
696:
691:
686:
683:
679:
675:
671:
667:
663:
658:
653:
649:
645:
640:
636:
632:
627:
626:
618:
615:
614:
608:
606:
602:
597:
595:
591:
587:
583:
579:
578:étale sheaves
575:
571:
567:
563:
560:
553:
547:
545:
541:
537:
533:
529:
524:
522:
518:
499:
496:
489:
485:
481:
476:
473:
467:
464:
459:
456:
446:
445:
444:
442:
438:
434:
430:
426:
422:
418:
414:
395:
392:
386:
383:
378:
375:
365:
364:
363:
360:
358:
354:
350:
339:
335:
331:
327:
323:
319:
315:
311:
308:− {0}). Here
307:
304:
300:
281:
276:
273:
268:
262:
259:
254:
251:
241:
240:
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231:
229:
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205:
204:local systems
201:
197:
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156:
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139:
135:
131:
127:
122:
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116:
112:
108:
104:
100:
96:
92:
88:
86:
83:Suppose that
76:
74:
70:
66:
62:
57:
55:
51:
47:
43:
39:
35:
30:
19:
1165:
1097:
1033:Riemann form
909:(1): 63–88,
906:
902:
865:
832:
803:
799:
776:
735:
702:
689:
647:
643:
634:
630:
604:
600:
598:
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543:
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535:
531:
527:
525:
516:
514:
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436:
432:
428:
424:
420:
416:
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410:
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356:
352:
348:
337:
333:
329:
325:
321:
317:
309:
305:
298:
296:
237:
227:
222:
221:In the case
220:
215:
209:
199:
190:
184:
178:
173:
161:
150:
146:
145:
142:
137:
133:
129:
125:
123:
118:
110:
106:
98:
94:
90:
89:
84:
82:
58:
28:
26:
1058:Riemann sum
1183:Categories
837:Birkhäuser
749:3540051902
657:1711.04148
631:Astérisque
623:References
682:119147066
477:−
158:D-modules
155:holonomic
79:Statement
42:monodromy
1167:Category
897:(1984),
831:(1991),
794:(1984),
734:(1970),
701:(1987),
611:See also
234:Examples
212:D-module
128:, where
915:0734785
888:0579742
855:1117227
822:0743382
785:0600704
758:0417174
725:0882000
674:3922360
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564:>0,
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886:
876:
853:
843:
820:
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766:169357
764:
756:
746:
723:
713:
680:
672:
109:. For
678:S2CID
652:arXiv
580:with
328:. If
164:with
126:Y − X
101:with
874:ISBN
841:ISBN
762:OCLC
744:ISBN
711:ISBN
191:and
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635:293
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576:of
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172:on
160:on
117:of
95:Sol
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818:MR
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742:,
721:MR
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670:MR
668:,
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210:A
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151:DR
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490:2
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