545:
388:
755:
237:
641:
1029:
399:
776:
Complex tori are special because they have multiple equivalent definitions of the Neron-Severi group. One definition uses its complex structure for the definition. For a complex torus
1057:
808:
1077:
868:
938:
1173:
La base per le varietà algebriche di dimensione qualunque contenute in una data e la teoria generale delle corrispondénze fra i punti di due superficie algebriche
311:
891:
652:
978:
958:
911:
848:
828:
143:
646:
and the Neron-Severi group can be identified with its image. Equivalently, by exactness, the Neron-Severi group is the kernel of the second arrow
262:). The elements of finite order are called Severi divisors, and form a finite group which is a birational invariant and whose order is called the
1117:
563:
986:
540:{\displaystyle \cdots \to H^{1}(V,{\mathcal {O}}_{V}^{*})\to H^{2}(V,2\pi i\mathbb {Z} )\to H^{2}(V,{\mathcal {O}}_{V})\to \cdots .}
120:
283:
1151:
275:
1188:
1146:
303:
33:
1159:
Problèmes arithmétiques et géometriques attachée à la notion de rang d'une courbe algébrique dans un corps
286:, the classification becomes amenable to discrete invariants. Algebraic equivalence is closely related to
1141:
287:
271:
29:
291:
1037:
779:
17:
1123:
1113:
1062:
853:
551:
49:
25:
761:
383:{\displaystyle 0\to 2\pi i\mathbb {Z} \to {\mathcal {O}}_{V}\to {\mathcal {O}}_{V}^{*}\to 0}
243:
61:
45:
916:
750:{\displaystyle \exp ^{*}\colon H^{2}(V,2\pi i\mathbb {Z} )\to H^{2}(V,{\mathcal {O}}_{V}).}
76:
72:
68:
873:
232:{\displaystyle 1\to \mathrm {Pic} ^{0}(V)\to \mathrm {Pic} (V)\to \mathrm {NS} (V)\to 0}
963:
943:
940:
makes it possible to identify the Neron-Severi group with the group of
Hermitian forms
896:
833:
813:
760:
In the complex case, the Neron-Severi group is therefore the group of 2-cocycles whose
135:
124:
1182:
1088:
41:
37:
765:
555:
131:
1112:(Second, augmented ed.). Berlin, Heidelberg: Springer Berlin Heidelberg.
1127:
282:; that is, using a stronger, non-linear equivalence relation in place of
636:{\displaystyle c_{1}\colon \mathrm {Pic} (V)\to H^{2}(V,\mathbb {Z} ),}
60:
In the cases of most importance to classical algebraic geometry, for a
1166:
La théorie de la base pour les diviseurs sur les variétés algébriques
1024:{\displaystyle {\text{Im}}H(\Lambda ,\Lambda )\subseteq \mathbb {Z} }
123:
by the Néron–Severi theorem, which was proved by Severi over the
1168:, Coll. Géom. Alg. Liège, G. Thone (1952) pp. 119–126
730:
514:
431:
358:
341:
1161:
Bull. Soc. Math. France, 80 (1952) pp. 101–166
1065:
1040:
989:
966:
946:
919:
899:
876:
856:
836:
816:
782:
764:
is represented by a complex hypersurface, that is, a
655:
566:
402:
314:
146:
1071:
1051:
1023:
972:
952:
932:
905:
885:
862:
842:
822:
802:
749:
635:
539:
382:
231:
298:First Chern class and integral valued 2-cocycles
1059:is an alternating integral form on the lattice
393:gives rise to a long exact sequence featuring
290:, an essentially topological classification by
1175:Mem. Accad. Ital., 5 (1934) pp. 239–283
1108:Birkenhake, Christina; Herbert Lange (2004).
8:
1064:
1041:
1039:
1017:
1016:
990:
988:
965:
945:
924:
918:
898:
875:
855:
835:
815:
792:
781:
735:
729:
728:
712:
698:
697:
673:
660:
654:
623:
622:
607:
580:
571:
565:
519:
513:
512:
496:
482:
481:
457:
441:
436:
430:
429:
413:
401:
368:
363:
357:
356:
346:
340:
339:
331:
330:
313:
206:
183:
165:
154:
145:
1100:
830:is a complex vector space of dimension
127:and by Néron over more general fields.
40:of a variety. Its rank is called the
7:
242:The fact that the rank is finite is
32:; in other words it is the group of
1066:
1007:
1001:
857:
797:
587:
584:
581:
210:
207:
190:
187:
184:
161:
158:
155:
14:
28:is the group of divisors modulo
121:finitely-generated abelian group
1010:
998:
741:
718:
705:
702:
679:
627:
613:
600:
597:
591:
528:
525:
502:
489:
486:
463:
450:
447:
419:
406:
374:
352:
335:
318:
284:linear equivalence of divisors
223:
220:
214:
203:
200:
194:
180:
177:
171:
150:
1:
1052:{\displaystyle {\text{Im}}H}
803:{\displaystyle X=V/\Lambda }
1147:Encyclopedia of Mathematics
75:of the Picard scheme is an
1205:
304:exponential sheaf sequence
1140:V.A. Iskovskikh (2001) ,
1110:Complex Abelian Varieties
1072:{\displaystyle \Lambda }
913:, the first Chern class
863:{\displaystyle \Lambda }
550:The first arrow is the
107:is an abelian group NS(
1073:
1053:
1032:
1025:
974:
954:
934:
907:
887:
864:
844:
824:
804:
751:
637:
541:
384:
233:
1074:
1054:
1026:
982:
975:
955:
935:
933:{\displaystyle c_{1}}
908:
888:
870:is a lattice of rank
865:
845:
825:
805:
752:
638:
542:
385:
288:numerical equivalence
272:algebraic equivalence
234:
44:. It is named after
30:algebraic equivalence
1142:"Néron–Severi group"
1063:
1038:
987:
964:
944:
917:
897:
874:
854:
834:
814:
780:
653:
564:
400:
312:
292:intersection numbers
144:
130:In other words, the
446:
373:
266:. Geometrically NS(
248:theorem of the base
73:connected component
1189:Algebraic geometry
1069:
1049:
1021:
970:
950:
930:
903:
886:{\displaystyle 2n}
883:
860:
840:
820:
800:
747:
633:
537:
428:
380:
355:
258:, often denoted ρ(
250:; the rank is the
229:
113:Néron–Severi group
22:Néron–Severi group
18:algebraic geometry
1119:978-3-662-06307-1
1044:
993:
973:{\displaystyle V}
953:{\displaystyle H}
906:{\displaystyle V}
843:{\displaystyle n}
823:{\displaystyle V}
552:first Chern class
1196:
1154:
1132:
1131:
1105:
1078:
1076:
1075:
1070:
1058:
1056:
1055:
1050:
1045:
1042:
1030:
1028:
1027:
1022:
1020:
994:
991:
979:
977:
976:
971:
959:
957:
956:
951:
939:
937:
936:
931:
929:
928:
912:
910:
909:
904:
892:
890:
889:
884:
869:
867:
866:
861:
849:
847:
846:
841:
829:
827:
826:
821:
809:
807:
806:
801:
796:
772:For complex tori
756:
754:
753:
748:
740:
739:
734:
733:
717:
716:
701:
678:
677:
665:
664:
642:
640:
639:
634:
626:
612:
611:
590:
576:
575:
546:
544:
543:
538:
524:
523:
518:
517:
501:
500:
485:
462:
461:
445:
440:
435:
434:
418:
417:
389:
387:
386:
381:
372:
367:
362:
361:
351:
350:
345:
344:
334:
270:) describes the
244:Francesco Severi
238:
236:
235:
230:
213:
193:
170:
169:
164:
62:complete variety
46:Francesco Severi
1204:
1203:
1199:
1198:
1197:
1195:
1194:
1193:
1179:
1178:
1139:
1136:
1135:
1120:
1107:
1106:
1102:
1097:
1085:
1061:
1060:
1036:
1035:
985:
984:
962:
961:
942:
941:
920:
915:
914:
895:
894:
872:
871:
852:
851:
832:
831:
812:
811:
778:
777:
774:
727:
708:
669:
656:
651:
650:
603:
567:
562:
561:
511:
492:
453:
409:
398:
397:
338:
310:
309:
300:
153:
142:
141:
125:complex numbers
77:abelian variety
58:
12:
11:
5:
1202:
1200:
1192:
1191:
1181:
1180:
1177:
1176:
1169:
1162:
1155:
1134:
1133:
1118:
1099:
1098:
1096:
1093:
1092:
1091:
1084:
1081:
1068:
1048:
1019:
1015:
1012:
1009:
1006:
1003:
1000:
997:
969:
949:
927:
923:
902:
882:
879:
859:
839:
819:
799:
795:
791:
788:
785:
773:
770:
758:
757:
746:
743:
738:
732:
726:
723:
720:
715:
711:
707:
704:
700:
696:
693:
690:
687:
684:
681:
676:
672:
668:
663:
659:
644:
643:
632:
629:
625:
621:
618:
615:
610:
606:
602:
599:
596:
593:
589:
586:
583:
579:
574:
570:
548:
547:
536:
533:
530:
527:
522:
516:
510:
507:
504:
499:
495:
491:
488:
484:
480:
477:
474:
471:
468:
465:
460:
456:
452:
449:
444:
439:
433:
427:
424:
421:
416:
412:
408:
405:
391:
390:
379:
376:
371:
366:
360:
354:
349:
343:
337:
333:
329:
326:
323:
320:
317:
299:
296:
240:
239:
228:
225:
222:
219:
216:
212:
209:
205:
202:
199:
196:
192:
189:
186:
182:
179:
176:
173:
168:
163:
160:
157:
152:
149:
136:exact sequence
111:), called the
105:
104:
89:
88:
57:
54:
13:
10:
9:
6:
4:
3:
2:
1201:
1190:
1187:
1186:
1184:
1174:
1171:F. Severi,
1170:
1167:
1163:
1160:
1156:
1153:
1149:
1148:
1143:
1138:
1137:
1129:
1125:
1121:
1115:
1111:
1104:
1101:
1094:
1090:
1089:Complex torus
1087:
1086:
1082:
1080:
1046:
1031:
1013:
1004:
995:
981:
967:
947:
925:
921:
900:
893:embedding in
880:
877:
837:
817:
793:
789:
786:
783:
771:
769:
767:
763:
762:Poincaré dual
744:
736:
724:
721:
713:
709:
694:
691:
688:
685:
682:
674:
670:
666:
661:
657:
649:
648:
647:
630:
619:
616:
608:
604:
594:
577:
572:
568:
560:
559:
558:
557:
553:
534:
531:
520:
508:
505:
497:
493:
478:
475:
472:
469:
466:
458:
454:
442:
437:
425:
422:
414:
410:
403:
396:
395:
394:
377:
369:
364:
347:
327:
324:
321:
315:
308:
307:
306:
305:
297:
295:
293:
289:
285:
281:
277:
273:
269:
265:
264:Severi number
261:
257:
253:
252:Picard number
249:
245:
226:
217:
197:
174:
166:
147:
140:
139:
138:
137:
134:fits into an
133:
128:
126:
122:
118:
114:
110:
102:
98:
94:
93:
92:
91:The quotient
86:
82:
81:
80:
78:
74:
70:
66:
63:
55:
53:
51:
47:
43:
42:Picard number
39:
38:Picard scheme
35:
31:
27:
23:
19:
1172:
1165:
1164:A. Néron,
1158:
1157:A. Néron,
1145:
1109:
1103:
1033:
983:
775:
766:Weil divisor
759:
645:
556:Picard group
549:
392:
301:
279:
267:
263:
259:
255:
251:
247:
241:
132:Picard group
129:
119:. This is a
116:
112:
108:
106:
100:
96:
90:
84:
69:non-singular
64:
59:
21:
15:
274:classes of
50:André Néron
1095:References
1034:Note that
56:Definition
34:components
1152:EMS Press
1128:851380558
1067:Λ
1014:⊆
1008:Λ
1002:Λ
980:such that
858:Λ
798:Λ
706:→
692:π
667::
662:∗
601:→
578::
532:⋯
529:→
490:→
476:π
451:→
443:∗
407:→
404:⋯
375:→
370:∗
353:→
336:→
325:π
319:→
224:→
204:→
181:→
151:→
1183:Category
1083:See also
810:, where
276:divisors
79:written
67:that is
554:on the
36:of the
26:variety
1126:
1116:
99:)/Pic(
71:, the
20:, the
24:of a
1124:OCLC
1114:ISBN
850:and
302:The
95:Pic(
83:Pic(
48:and
960:on
658:exp
278:on
254:of
246:'s
115:of
16:In
1185::
1150:,
1144:,
1122:.
1079:.
1043:Im
992:Im
768:.
294:.
87:).
52:.
1130:.
1047:H
1018:Z
1011:)
1005:,
999:(
996:H
968:V
948:H
926:1
922:c
901:V
881:n
878:2
838:n
818:V
794:/
790:V
787:=
784:X
745:.
742:)
737:V
731:O
725:,
722:V
719:(
714:2
710:H
703:)
699:Z
695:i
689:2
686:,
683:V
680:(
675:2
671:H
631:,
628:)
624:Z
620:,
617:V
614:(
609:2
605:H
598:)
595:V
592:(
588:c
585:i
582:P
573:1
569:c
535:.
526:)
521:V
515:O
509:,
506:V
503:(
498:2
494:H
487:)
483:Z
479:i
473:2
470:,
467:V
464:(
459:2
455:H
448:)
438:V
432:O
426:,
423:V
420:(
415:1
411:H
378:0
365:V
359:O
348:V
342:O
332:Z
328:i
322:2
316:0
280:V
268:V
260:V
256:V
227:0
221:)
218:V
215:(
211:S
208:N
201:)
198:V
195:(
191:c
188:i
185:P
178:)
175:V
172:(
167:0
162:c
159:i
156:P
148:1
117:V
109:V
103:)
101:V
97:V
85:V
65:V
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