25:
1013:
of solutions modulo gauge, and the index of the derivative is the virtual dimension. The localized Euler class of the pair (E,s) is a homology class with closed support on the zero set of the section. Its dimension is the index of the derivative. When the section is transversal, the class is just the fundamental class of the zero set with the proper orientation. The class is well behaved in one parameter families and therefore defines the “right” fundamental cycle even if the section is no longer transversal.
1026:
1012:
Consider an infinite dimensional bundle E over an infinite dimensional manifold M with a section s with
Fredholm derivative. In practice this situation occurs whenever we have system of PDE’s which are elliptic when considered modulo some gauge group action. The zero set Z(s) is then the moduli space
1056:
This formula enables us to compute the conductor that measures the wild ramification by using the sheaf of differential 1-forms. S. Bloch conjectures a formula for the Artin conductor of the â„“-adic etale cohomology of a regular model of a variety over a local field and proves it for a curve. The
698:
533:
259:
773:
1090:
S. Bloch, “Cycles on arithmetic schemes and Euler characteristics of curves,” Algebraic geometry, Bowdoin, 1985, 421–450, Proc. Symp. Pure Math. 46, Part 2, Amer. Math. Soc., Providence, RI, 1987.
931:
591:
408:
454:
1003:
157:
318:
857:
345:
813:
793:
961:
375:
288:
1061:
599:
1057:
deepest result about the Bloch conductor is its equality with the Artin conductor, defined in terms of the l-adic cohomology of X, in certain cases.
459:
1110:
46:
1101:, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics , vol. 2, Berlin, New York:
168:
706:
89:, that is defined for a chain complex of vector bundles as opposed to a single vector bundle. It was originally introduced in Fulton's
113:
68:
1094:
98:
1140:
873:
39:
33:
50:
542:
380:
413:
970:
816:
291:
135:
94:
297:
128:
be a pure-dimensional regular scheme of finite type over a field or discrete valuation ring and
1127:
K. Kato and T. Saito, “On the conductor formula of Bloch,” Publ. Math. IHÉS 100 (2005), 5-151.
1106:
830:
323:
105:
798:
778:
348:
1120:
939:
353:
1116:
1102:
267:
693:{\displaystyle c_{i,X}^{Y}(E_{\bullet })\cap \alpha =\eta _{*}(c_{i}(\xi )\cap \gamma )}
1025:
1134:
867:
is smooth over a field, then the localized Chern class coincides with the class
86:
528:{\displaystyle \xi =\prod (-1)^{i}\operatorname {pr} _{i}^{*}(\xi _{i})}
254:{\displaystyle 0=E_{n-1}\to E_{n}\to \dots \to E_{m}\to E_{m-1}=0}
768:{\displaystyle \eta :G_{n}\times _{Y}\dots \times _{Y}G_{m}\to X}
1020:
290:. The localized Chern class of this complex is a class in the
18:
112:
that computes the non-constancy of Euler characteristic of a
93:, as an algebraic counterpart of the similar construction in
116:
of algebraic varieties (in the mixed characteristic case).
108:(schemes over a Dedekind domain) for the purpose of giving
104:
S. Bloch later generalized the notion in the context of
1037:
973:
942:
876:
833:
801:
781:
709:
602:
545:
462:
416:
383:
356:
326:
300:
270:
171:
138:
997:
955:
925:
851:
807:
787:
767:
692:
585:
527:
448:
402:
369:
339:
312:
282:
253:
151:
109:
963:is the section determined by the differential of
926:{\displaystyle (-1)^{\dim X}\mathbf {Z} (s_{f})}
8:
97:. The notion is used in particular in the
986:
974:
972:
947:
941:
914:
902:
890:
875:
832:
800:
780:
753:
743:
730:
720:
708:
666:
653:
631:
618:
607:
601:
586:{\displaystyle c_{i,X}^{Y}(E_{\bullet })}
574:
561:
550:
544:
516:
500:
495:
485:
461:
434:
421:
415:
394:
382:
361:
355:
331:
325:
299:
269:
233:
220:
201:
182:
170:
143:
137:
69:Learn how and when to remove this message
403:{\displaystyle \operatorname {rk} E_{i}}
32:This article includes a list of general
1072:
1079:
1005:is the class of the singular locus of
347:denote the tautological bundle of the
159:denote a complex of vector bundles on
7:
1062:Grothendieck–Ogg–Shafarevich formula
449:{\displaystyle E_{i}\otimes E_{i-1}}
998:{\displaystyle \mathbf {Z} (s_{f})}
38:it lacks sufficient corresponding
14:
1016:
1024:
975:
903:
860:
23:
992:
979:
920:
907:
887:
877:
843:
823:Example: localized Euler class
759:
687:
678:
672:
659:
637:
624:
580:
567:
522:
509:
482:
472:
226:
213:
207:
194:
1:
152:{\displaystyle E_{\bullet }}
593:is defined by the formula:
1157:
1059:
539:-th localized Chern class
313:{\displaystyle X\subset Y}
110:#Bloch's conductor formula
1017:Bloch's conductor formula
795:is a cycle obtained from
99:Riemann–Roch-type theorem
81:In algebraic geometry, a
852:{\displaystyle f:X\to S}
340:{\displaystyle \xi _{i}}
320:defined as follows. Let
132:a closed subscheme. Let
808:{\displaystyle \alpha }
788:{\displaystyle \gamma }
53:more precise citations.
999:
957:
927:
853:
809:
789:
775:is the projection and
769:
694:
587:
529:
450:
404:
371:
341:
314:
284:
255:
153:
1000:
958:
956:{\displaystyle s_{f}}
928:
854:
810:
790:
770:
695:
588:
530:
451:
405:
372:
370:{\displaystyle G_{i}}
342:
315:
285:
256:
154:
83:localized Chern class
971:
940:
874:
831:
799:
779:
707:
600:
543:
460:
414:
381:
354:
324:
298:
292:bivariant Chow group
268:
169:
136:
1099:Intersection theory
623:
566:
505:
283:{\displaystyle Y-X}
114:degenerating family
91:intersection theory
16:Concept in geometry
1141:Algebraic geometry
1036:. You can help by
995:
953:
923:
849:
817:graph construction
805:
785:
765:
690:
603:
583:
546:
525:
491:
446:
400:
367:
337:
310:
280:
251:
149:
106:arithmetic schemes
95:algebraic topology
85:is a variant of a
1112:978-3-540-62046-4
1082:, Example 18.1.3.
1054:
1053:
815:by the so-called
264:that is exact on
79:
78:
71:
1148:
1123:
1083:
1077:
1049:
1046:
1028:
1021:
1004:
1002:
1001:
996:
991:
990:
978:
962:
960:
959:
954:
952:
951:
936:where, roughly,
932:
930:
929:
924:
919:
918:
906:
901:
900:
858:
856:
855:
850:
814:
812:
811:
806:
794:
792:
791:
786:
774:
772:
771:
766:
758:
757:
748:
747:
735:
734:
725:
724:
699:
697:
696:
691:
671:
670:
658:
657:
636:
635:
622:
617:
592:
590:
589:
584:
579:
578:
565:
560:
534:
532:
531:
526:
521:
520:
504:
499:
490:
489:
455:
453:
452:
447:
445:
444:
426:
425:
409:
407:
406:
401:
399:
398:
376:
374:
373:
368:
366:
365:
349:Grassmann bundle
346:
344:
343:
338:
336:
335:
319:
317:
316:
311:
289:
287:
286:
281:
260:
258:
257:
252:
244:
243:
225:
224:
206:
205:
193:
192:
158:
156:
155:
150:
148:
147:
74:
67:
63:
60:
54:
49:this article by
40:inline citations
27:
26:
19:
1156:
1155:
1151:
1150:
1149:
1147:
1146:
1145:
1131:
1130:
1113:
1103:Springer-Verlag
1095:Fulton, William
1093:
1087:
1086:
1078:
1074:
1069:
1064:
1050:
1044:
1041:
1034:needs expansion
1019:
982:
969:
968:
943:
938:
937:
910:
886:
872:
871:
829:
828:
825:
797:
796:
777:
776:
749:
739:
726:
716:
705:
704:
662:
649:
627:
598:
597:
570:
541:
540:
512:
481:
458:
457:
430:
417:
412:
411:
410:sub-bundles of
390:
379:
378:
357:
352:
351:
327:
322:
321:
296:
295:
266:
265:
229:
216:
197:
178:
167:
166:
139:
134:
133:
122:
75:
64:
58:
55:
45:Please help to
44:
28:
24:
17:
12:
11:
5:
1154:
1152:
1144:
1143:
1133:
1132:
1129:
1128:
1125:
1111:
1091:
1085:
1084:
1071:
1070:
1068:
1065:
1052:
1051:
1031:
1029:
1018:
1015:
994:
989:
985:
981:
977:
950:
946:
934:
933:
922:
917:
913:
909:
905:
899:
896:
893:
889:
885:
882:
879:
848:
845:
842:
839:
836:
824:
821:
804:
784:
764:
761:
756:
752:
746:
742:
738:
733:
729:
723:
719:
715:
712:
701:
700:
689:
686:
683:
680:
677:
674:
669:
665:
661:
656:
652:
648:
645:
642:
639:
634:
630:
626:
621:
616:
613:
610:
606:
582:
577:
573:
569:
564:
559:
556:
553:
549:
524:
519:
515:
511:
508:
503:
498:
494:
488:
484:
480:
477:
474:
471:
468:
465:
443:
440:
437:
433:
429:
424:
420:
397:
393:
389:
386:
364:
360:
334:
330:
309:
306:
303:
279:
276:
273:
262:
261:
250:
247:
242:
239:
236:
232:
228:
223:
219:
215:
212:
209:
204:
200:
196:
191:
188:
185:
181:
177:
174:
146:
142:
121:
118:
77:
76:
31:
29:
22:
15:
13:
10:
9:
6:
4:
3:
2:
1153:
1142:
1139:
1138:
1136:
1126:
1124:, section B.7
1122:
1118:
1114:
1108:
1104:
1100:
1096:
1092:
1089:
1088:
1081:
1076:
1073:
1066:
1063:
1058:
1048:
1045:November 2019
1039:
1035:
1032:This section
1030:
1027:
1023:
1022:
1014:
1010:
1008:
987:
983:
966:
948:
944:
915:
911:
897:
894:
891:
883:
880:
870:
869:
868:
866:
862:
846:
840:
837:
834:
822:
820:
818:
802:
782:
762:
754:
750:
744:
740:
736:
731:
727:
721:
717:
713:
710:
684:
681:
675:
667:
663:
654:
650:
646:
643:
640:
632:
628:
619:
614:
611:
608:
604:
596:
595:
594:
575:
571:
562:
557:
554:
551:
547:
538:
517:
513:
506:
501:
496:
492:
486:
478:
475:
469:
466:
463:
441:
438:
435:
431:
427:
422:
418:
395:
391:
387:
384:
362:
358:
350:
332:
328:
307:
304:
301:
293:
277:
274:
271:
248:
245:
240:
237:
234:
230:
221:
217:
210:
202:
198:
189:
186:
183:
179:
175:
172:
165:
164:
163:
162:
144:
140:
131:
127:
119:
117:
115:
111:
107:
102:
100:
96:
92:
88:
84:
73:
70:
62:
59:November 2019
52:
48:
42:
41:
35:
30:
21:
20:
1098:
1075:
1055:
1042:
1038:adding to it
1033:
1011:
1006:
964:
935:
864:
861:#Definitions
826:
702:
536:
263:
160:
129:
125:
123:
103:
90:
82:
80:
65:
56:
37:
1080:Fulton 1998
967:and (thus)
535:. Then the
120:Definitions
87:Chern class
51:introducing
1067:References
1060:See also:
34:references
895:
881:−
859:be as in
844:→
803:α
783:γ
760:→
741:×
737:⋯
728:×
711:η
685:γ
682:∩
676:ξ
655:∗
651:η
644:α
641:∩
633:∙
576:∙
514:ξ
507:
502:∗
476:−
470:∏
464:ξ
439:−
428:⊗
388:
329:ξ
305:⊂
275:−
238:−
227:→
214:→
211:⋯
208:→
195:→
187:−
145:∙
1135:Category
1097:(1998),
377:of rank
1121:1644323
47:improve
1119:
1109:
703:where
456:. Let
36:, but
863:. If
1107:ISBN
827:Let
124:Let
1040:.
892:dim
294:of
1137::
1117:MR
1115:,
1105:,
1009:.
819:.
493:pr
385:rk
101:.
1047:)
1043:(
1007:f
993:)
988:f
984:s
980:(
976:Z
965:f
949:f
945:s
921:)
916:f
912:s
908:(
904:Z
898:X
888:)
884:1
878:(
865:S
847:S
841:X
838::
835:f
763:X
755:m
751:G
745:Y
732:Y
722:n
718:G
714::
688:)
679:)
673:(
668:i
664:c
660:(
647:=
638:)
629:E
625:(
620:Y
615:X
612:,
609:i
605:c
581:)
572:E
568:(
563:Y
558:X
555:,
552:i
548:c
537:i
523:)
518:i
510:(
497:i
487:i
483:)
479:1
473:(
467:=
442:1
436:i
432:E
423:i
419:E
396:i
392:E
363:i
359:G
333:i
308:Y
302:X
278:X
272:Y
249:0
246:=
241:1
235:m
231:E
222:m
218:E
203:n
199:E
190:1
184:n
180:E
176:=
173:0
161:Y
141:E
130:X
126:Y
72:)
66:(
61:)
57:(
43:.
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
