422:
822:
148:
1175:
557:
1323:
1237:
880:
1123:
662:
1441:
L vanishing theorems for positive line bundles and adjunction theory, Lecture Notes of a CIME course on "Transcendental
Methods of Algebraic Geometry" (Cetraro, Italy, July 1994)
1051:
982:
952:
724:
272:
308:
494:
449:
210:
914:
612:
328:
168:
1265:
1005:
685:
580:
333:
757:
1505:
73:
1132:
1464:
1413:
1456:
499:
1286:
1186:
1541:
1536:
837:
1531:
1489:
1401:
1016:
1084:
617:
1243:
if it is a linear combination of products of semi-positive forms, with positive real coefficients. A real
17:
1026:
957:
927:
690:
222:
1435:
889:
277:
1328:
Weakly positive and strongly positive forms form convex cones. On compact manifolds these cones are
457:
36:
427:
1501:
1460:
1448:
1409:
1397:
1020:
740:
192:
32:
899:
588:
313:
153:
1493:
1424:
727:
24:
1515:
1474:
1250:
987:
667:
562:
1511:
1470:
1078:
831:
is equipped with a unique connection preserving the
Hermitian structure and satisfying
213:
1525:
1481:
417:{\displaystyle \omega ={\sqrt {-1}}\sum _{i}\alpha _{i}dz_{i}\wedge d{\bar {z}}_{i},}
212:
is the imaginary part of a positive semidefinite (respectively, positive definite)
739:
In algebraic geometry, positive definite (1,1)-forms arise as curvature forms of
1329:
1126:
1070:
1008:
916:
of the Chern connection is always a purely imaginary (1,1)-form. A line bundle
1497:
1428:
1420:
1440:
817:{\displaystyle {\bar {\partial }}:\;L\mapsto L\otimes \Lambda ^{0,1}(M)}
143:{\displaystyle \Lambda ^{p,p}(M)\cap \Lambda ^{2p}(M,{\mathbb {R} }).}
954:
is a positive (1,1)-form. (Note that the de Rham cohomology class of
1239:, there are two different notions of positivity. A form is called
1019:
claims that a positive line bundle is ample, and conversely, any
1170:{\displaystyle \eta ,\zeta \mapsto \int _{M}\eta \wedge \zeta }
751:
be a holomorphic
Hermitian line bundle on a complex manifold,
552:{\displaystyle -{\sqrt {-1}}\omega (v,{\bar {v}})\geq 0}
186:) if any of the following equivalent conditions holds:
1486:
Hodge Theory and
Complex Algebraic Geometry (2 vols.)
1289:
1253:
1189:
1135:
1087:
1029:
990:
960:
930:
902:
840:
760:
693:
670:
620:
591:
565:
502:
460:
430:
336:
316:
280:
225:
195:
156:
76:
1317:
1259:
1231:
1169:
1117:
1045:
999:
976:
946:
908:
874:
816:
718:
679:
656:
606:
574:
551:
488:
443:
416:
322:
302:
266:
204:
162:
142:
1318:{\displaystyle \int _{M}\eta \wedge \zeta \geq 0}
1232:{\displaystyle 2\leq p\leq dim_{\mathbb {C} }M-2}
875:{\displaystyle \nabla ^{0,1}={\bar {\partial }}}
451:real and non-negative (respectively, positive).
1129:, with respect to the Poincaré pairing :
70:) and real, that is, lie in the intersection
8:
776:
700:
1294:
1288:
1252:
1214:
1213:
1212:
1188:
1152:
1134:
1100:
1099:
1098:
1086:
1030:
1028:
989:
961:
959:
931:
929:
901:
861:
860:
845:
839:
793:
762:
761:
759:
692:
669:
619:
590:
564:
529:
528:
506:
501:
471:
459:
435:
429:
405:
394:
393:
380:
367:
357:
343:
335:
315:
285:
279:
258:
233:
224:
194:
155:
129:
128:
127:
109:
81:
75:
1341:
1283:-forms ζ with compact support, we have
1332:with respect to the Poincaré pairing.
1419:Griffiths, Phillip (3 January 2020).
1118:{\displaystyle dim_{\mathbb {C} }M=2}
827:its complex structure operator. Then
657:{\displaystyle \omega (v,I(v))\geq 0}
7:
1421:"Positivity and Vanishing Theorems"
1046:{\displaystyle {\sqrt {-1}}\Theta }
977:{\displaystyle {\sqrt {-1}}\Theta }
947:{\displaystyle {\sqrt {-1}}\Theta }
1040:
971:
941:
903:
863:
842:
790:
764:
282:
106:
78:
31:refers to several classes of real
14:
1453:Complex Geometry: An Introduction
1406:Principles of Algebraic Geometry
719:{\displaystyle I:\;TM\mapsto TM}
267:{\displaystyle dz_{1},...dz_{n}}
1023:admits a Hermitian metric with
303:{\displaystyle \Lambda ^{1,0}M}
1271:-dimensional complex manifold
1145:
866:
811:
805:
780:
767:
707:
645:
642:
636:
624:
540:
534:
519:
399:
330:can be written diagonally, as
134:
118:
99:
93:
58:)-forms on a complex manifold
16:For the linguistics term, see
1:
1279:if for all strongly positive
1065:Semi-positive (1,1)-forms on
489:{\displaystyle v\in T^{1,0}M}
454:For any (1,0)-tangent vector
62:are forms which are of type (
585:For any real tangent vector
444:{\displaystyle \alpha _{i}}
1558:
1490:Cambridge University Press
886:This connection is called
15:
1017:Kodaira embedding theorem
1498:10.1017/CBO9780511615344
205:{\displaystyle -\omega }
909:{\displaystyle \Theta }
607:{\displaystyle v\in TM}
323:{\displaystyle \omega }
163:{\displaystyle \omega }
1319:
1261:
1233:
1171:
1119:
1047:
1001:
978:
948:
910:
876:
818:
720:
681:
658:
608:
576:
553:
490:
445:
418:
324:
304:
268:
206:
164:
144:
18:Positive (linguistics)
1320:
1262:
1260:{\displaystyle \eta }
1234:
1172:
1120:
1048:
1002:
1000:{\displaystyle 2\pi }
979:
949:
911:
877:
819:
745:positive line bundles
735:Positive line bundles
721:
682:
680:{\displaystyle >0}
659:
609:
577:
575:{\displaystyle >0}
554:
491:
446:
419:
325:
305:
269:
207:
165:
145:
1287:
1251:
1187:
1133:
1085:
1027:
988:
958:
928:
900:
838:
758:
691:
668:
618:
589:
563:
500:
458:
428:
334:
314:
278:
223:
193:
154:
74:
1542:Differential forms
1537:Algebraic geometry
1449:Huybrechts, Daniel
1315:
1257:
1229:
1167:
1115:
1043:
997:
974:
944:
906:
872:
814:
741:ample line bundles
716:
677:
654:
604:
572:
549:
486:
441:
414:
362:
320:
300:
264:
202:
160:
150:A real (1,1)-form
140:
33:differential forms
1532:Complex manifolds
1507:978-0-521-71801-1
1429:20.500.12111/7881
1366:Huybrechts (2005)
1348:Huybrechts (2005)
1241:strongly positive
1038:
1021:ample line bundle
969:
939:
869:
770:
728:complex structure
537:
514:
402:
353:
351:
184:positive definite
178:), respectively,
1549:
1518:
1477:
1432:
1385:
1382:
1376:
1373:
1367:
1364:
1358:
1355:
1349:
1346:
1324:
1322:
1321:
1316:
1299:
1298:
1266:
1264:
1263:
1258:
1238:
1236:
1235:
1230:
1219:
1218:
1217:
1176:
1174:
1173:
1168:
1157:
1156:
1124:
1122:
1121:
1116:
1105:
1104:
1103:
1052:
1050:
1049:
1044:
1039:
1031:
1007:times the first
1006:
1004:
1003:
998:
983:
981:
980:
975:
970:
962:
953:
951:
950:
945:
940:
932:
915:
913:
912:
907:
890:Chern connection
881:
879:
878:
873:
871:
870:
862:
856:
855:
823:
821:
820:
815:
804:
803:
772:
771:
763:
725:
723:
722:
717:
686:
684:
683:
678:
663:
661:
660:
655:
613:
611:
610:
605:
581:
579:
578:
573:
558:
556:
555:
550:
539:
538:
530:
515:
507:
495:
493:
492:
487:
482:
481:
450:
448:
447:
442:
440:
439:
423:
421:
420:
415:
410:
409:
404:
403:
395:
385:
384:
372:
371:
361:
352:
344:
329:
327:
326:
321:
310:of (1,0)-forms,
309:
307:
306:
301:
296:
295:
273:
271:
270:
265:
263:
262:
238:
237:
211:
209:
208:
203:
174:(sometimes just
169:
167:
166:
161:
149:
147:
146:
141:
133:
132:
117:
116:
92:
91:
25:complex geometry
1557:
1556:
1552:
1551:
1550:
1548:
1547:
1546:
1522:
1521:
1508:
1480:
1467:
1447:
1418:
1394:
1389:
1388:
1384:Demailly (1994)
1383:
1379:
1375:Demailly (1994)
1374:
1370:
1365:
1361:
1357:Demailly (1994)
1356:
1352:
1347:
1343:
1338:
1290:
1285:
1284:
1277:weakly positive
1249:
1248:
1208:
1185:
1184:
1148:
1131:
1130:
1125:, this cone is
1094:
1083:
1082:
1079:complex surface
1063:
1057:Positivity for
1025:
1024:
986:
985:
956:
955:
926:
925:
898:
897:
841:
836:
835:
789:
756:
755:
743:(also known as
737:
689:
688:
666:
665:
664:(respectively,
616:
615:
587:
586:
561:
560:
559:(respectively,
498:
497:
467:
456:
455:
431:
426:
425:
392:
376:
363:
332:
331:
312:
311:
281:
276:
275:
254:
229:
221:
220:
219:For some basis
191:
190:
152:
151:
105:
77:
72:
71:
48:
21:
12:
11:
5:
1555:
1553:
1545:
1544:
1539:
1534:
1524:
1523:
1520:
1519:
1506:
1482:Voisin, Claire
1478:
1465:
1445:
1436:J.-P. Demailly
1433:
1416:
1393:
1390:
1387:
1386:
1377:
1368:
1359:
1350:
1340:
1339:
1337:
1334:
1314:
1311:
1308:
1305:
1302:
1297:
1293:
1256:
1228:
1225:
1222:
1216:
1211:
1207:
1204:
1201:
1198:
1195:
1192:
1183:-forms, where
1166:
1163:
1160:
1155:
1151:
1147:
1144:
1141:
1138:
1114:
1111:
1108:
1102:
1097:
1093:
1090:
1062:
1055:
1042:
1037:
1034:
996:
993:
973:
968:
965:
943:
938:
935:
905:
896:The curvature
884:
883:
868:
865:
859:
854:
851:
848:
844:
825:
824:
813:
810:
807:
802:
799:
796:
792:
788:
785:
782:
779:
775:
769:
766:
736:
733:
732:
731:
715:
712:
709:
706:
703:
699:
696:
676:
673:
653:
650:
647:
644:
641:
638:
635:
632:
629:
626:
623:
603:
600:
597:
594:
583:
571:
568:
548:
545:
542:
536:
533:
527:
524:
521:
518:
513:
510:
505:
485:
480:
477:
474:
470:
466:
463:
452:
438:
434:
413:
408:
401:
398:
391:
388:
383:
379:
375:
370:
366:
360:
356:
350:
347:
342:
339:
319:
299:
294:
291:
288:
284:
261:
257:
253:
250:
247:
244:
241:
236:
232:
228:
217:
214:Hermitian form
201:
198:
159:
139:
136:
131:
126:
123:
120:
115:
112:
108:
104:
101:
98:
95:
90:
87:
84:
80:
47:
44:
13:
10:
9:
6:
4:
3:
2:
1554:
1543:
1540:
1538:
1535:
1533:
1530:
1529:
1527:
1517:
1513:
1509:
1503:
1499:
1495:
1491:
1487:
1483:
1479:
1476:
1472:
1468:
1466:3-540-21290-6
1462:
1458:
1454:
1450:
1446:
1443:
1442:
1437:
1434:
1430:
1426:
1422:
1417:
1415:
1414:0-471-32792-1
1411:
1407:
1403:
1399:
1396:
1395:
1391:
1381:
1378:
1372:
1369:
1363:
1360:
1354:
1351:
1345:
1342:
1335:
1333:
1331:
1326:
1312:
1309:
1306:
1303:
1300:
1295:
1291:
1282:
1278:
1274:
1270:
1254:
1246:
1242:
1226:
1223:
1220:
1209:
1205:
1202:
1199:
1196:
1193:
1190:
1182:
1177:
1164:
1161:
1158:
1153:
1149:
1142:
1139:
1136:
1128:
1112:
1109:
1106:
1095:
1091:
1088:
1080:
1077:is a compact
1076:
1072:
1068:
1060:
1056:
1054:
1035:
1032:
1022:
1018:
1014:
1010:
994:
991:
966:
963:
936:
933:
923:
919:
894:
892:
891:
857:
852:
849:
846:
834:
833:
832:
830:
808:
800:
797:
794:
786:
783:
777:
773:
754:
753:
752:
750:
746:
742:
734:
729:
713:
710:
704:
701:
697:
694:
674:
671:
651:
648:
639:
633:
630:
627:
621:
601:
598:
595:
592:
584:
569:
566:
546:
543:
531:
525:
522:
516:
511:
508:
503:
483:
478:
475:
472:
468:
464:
461:
453:
436:
432:
411:
406:
396:
389:
386:
381:
377:
373:
368:
364:
358:
354:
348:
345:
340:
337:
317:
297:
292:
289:
286:
274:in the space
259:
255:
251:
248:
245:
242:
239:
234:
230:
226:
218:
215:
199:
196:
189:
188:
187:
185:
181:
177:
173:
172:semi-positive
157:
137:
124:
121:
113:
110:
102:
96:
88:
85:
82:
69:
65:
61:
57:
53:
45:
43:
41:
38:
34:
30:
29:positive form
26:
19:
1485:
1452:
1439:
1405:
1398:P. Griffiths
1380:
1371:
1362:
1353:
1344:
1327:
1280:
1276:
1272:
1268:
1244:
1240:
1180:
1178:
1074:
1066:
1064:
1058:
1012:
921:
917:
895:
887:
885:
828:
826:
748:
744:
738:
183:
179:
175:
171:
67:
63:
59:
55:
51:
49:
39:
28:
22:
1071:convex cone
1009:Chern class
46:(1,1)-forms
27:, the term
1526:Categories
1392:References
1281:(n-p, n-p)
1275:is called
1053:positive.
920:is called
170:is called
37:Hodge type
1484:(2007) ,
1408:, Wiley.
1402:J. Harris
1310:≥
1307:ζ
1304:∧
1301:η
1292:∫
1255:η
1224:−
1200:≤
1194:≤
1165:ζ
1162:∧
1159:η
1150:∫
1146:↦
1143:ζ
1137:η
1127:self-dual
1041:Θ
1033:−
995:π
972:Θ
964:−
942:Θ
934:−
904:Θ
867:¯
864:∂
843:∇
791:Λ
787:⊗
781:↦
768:¯
765:∂
730:operator.
708:↦
687:), where
649:≥
622:ω
596:∈
544:≥
535:¯
517:ω
509:−
504:−
465:∈
433:α
400:¯
387:∧
365:α
355:∑
346:−
338:ω
318:ω
283:Λ
200:ω
197:−
158:ω
107:Λ
103:∩
79:Λ
1457:Springer
1451:(2005),
1404:(1978),
922:positive
180:positive
176:positive
1516:1967689
1475:2093043
1073:. When
1069:form a
1015:.) The
747:). Let
726:is the
1514:
1504:
1473:
1463:
1412:
1267:on an
1247:-form
1245:(p, p)
1181:(p, p)
1061:-forms
1059:(p, p)
50:Real (
40:(p, p)
1336:Notes
424:with
1502:ISBN
1461:ISBN
1410:ISBN
1400:and
1330:dual
1179:For
888:the
672:>
567:>
182:(or
1494:doi
1425:hdl
1011:of
984:is
924:if
35:of
23:In
1528::
1512:MR
1510:,
1500:,
1492:,
1488:,
1471:MR
1469:,
1459:,
1455:,
1438:,
1423:.
1325:.
1081:,
893:.
614:,
582:).
496:,
42:.
1496::
1444:.
1431:.
1427::
1313:0
1296:M
1273:M
1269:n
1227:2
1221:M
1215:C
1210:m
1206:i
1203:d
1197:p
1191:2
1154:M
1140:,
1113:2
1110:=
1107:M
1101:C
1096:m
1092:i
1089:d
1075:M
1067:M
1036:1
1013:L
992:2
967:1
937:1
918:L
882:.
858:=
853:1
850:,
847:0
829:L
812:)
809:M
806:(
801:1
798:,
795:0
784:L
778:L
774::
749:L
714:M
711:T
705:M
702:T
698::
695:I
675:0
652:0
646:)
643:)
640:v
637:(
634:I
631:,
628:v
625:(
602:M
599:T
593:v
570:0
547:0
541:)
532:v
526:,
523:v
520:(
512:1
484:M
479:0
476:,
473:1
469:T
462:v
437:i
412:,
407:i
397:z
390:d
382:i
378:z
374:d
369:i
359:i
349:1
341:=
298:M
293:0
290:,
287:1
260:n
256:z
252:d
249:.
246:.
243:.
240:,
235:1
231:z
227:d
216:.
138:.
135:)
130:R
125:,
122:M
119:(
114:p
111:2
100:)
97:M
94:(
89:p
86:,
83:p
68:p
66:,
64:p
60:M
56:p
54:,
52:p
20:.
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