1632:. The fact that there are nilmanifolds that are not diffeomorphic to a torus shows that there is some space between Kähler manifolds and symplectic manifolds, but the class of nilmanifolds fails to show any differences between Kähler manifolds, Lefschetz manifolds, and strong Lefschetz manifolds.
1643:. This has been proved. Furthermore, many examples have been found of solvmanifolds that are strong Lefschetz but not Kähler, and solvmanifolds that are Lefschetz but not strong Lefschetz. Such examples were given by Takumi Yamada in 2002.
234:
773:
1278:
1030:
1128:
501:
123:
358:
67:
908:
858:
1313:
657:
1503:
1441:
1382:
1665:
1612:
1592:
1572:
572:
625:
1405:
299:
796:
592:
385:
276:
1533:
1471:
1339:
1157:
534:
910:
are vector spaces of the same dimension, so in these cases it is natural to ask whether or not the various iterations of
Lefschetz maps are isomorphisms.
131:
1551:
tells us that it is also a strong
Lefschetz manifold, and hence a Lefschetz manifold. Therefore we have the following chain of inclusions.
246:
are even. This remark leads to numerous examples of symplectic manifolds which are not Kähler, the first historical example is due to
665:
1173:
931:
1041:
393:
1802:
1548:
914:
84:
74:
1636:
307:
30:
863:
813:
1408:
1286:
807:
630:
25:
1718:
1682:
1476:
1414:
1355:
364:
1544:
70:
1709:
1618:
1597:
1577:
1557:
539:
1768:
1727:
1674:
1660:
1622:
247:
1782:
1741:
1694:
1778:
1737:
1690:
597:
242:
The topology of these symplectic manifolds is severely constrained, for example their odd
1387:
281:
781:
577:
370:
261:
1512:
1450:
1318:
1136:
513:
1796:
1732:
1713:
799:
917:
states that this is the case for the symplectic form of a compact Kähler manifold.
243:
1756:
1625:
17:
229:{\displaystyle \cup \colon H^{n-k}(M,\mathbb {R} )\to H^{n+k}(M,\mathbb {R} )}
1773:
803:
1686:
1678:
1640:
1629:
1411:. Then it is orientable, but maybe not compact. One says that
768:{\displaystyle L_{}^{i}:H_{DR}^{n-i}(M)\to H_{DR}^{n+i}(M).}
1273:{\displaystyle L_{}^{i}:H_{DR}^{n-i}(M)\to H_{DR}^{n+i}(M)}
1639:
admits a Kähler structure, then it is diffeomorphic to a
1025:{\displaystyle L_{}^{n-1}:H_{DR}^{1}(M)\to H_{DR}^{2n-1}}
1663:(1976). "Some simple examples of symplectic manifolds".
1628:
is a
Lefschetz manifold, then it is diffeomorphic to a
1600:
1580:
1560:
1515:
1479:
1453:
1417:
1390:
1358:
1321:
1289:
1176:
1139:
1123:{\displaystyle L_{}^{n}:H_{DR}^{0}(M)\to H_{DR}^{2n}}
1044:
934:
866:
816:
784:
668:
633:
600:
580:
542:
516:
396:
373:
310:
284:
264:
134:
87:
33:
496:{\displaystyle L_{}:H_{DR}(M)\to H_{DR}(M),\mapsto }
1714:"Kähler and symplectic structures on nilmanifolds"
1606:
1586:
1566:
1527:
1497:
1465:
1435:
1399:
1376:
1333:
1307:
1272:
1151:
1122:
1024:
902:
852:
790:
767:
651:
619:
586:
566:
528:
495:
379:
352:
293:
270:
228:
117:
61:
1666:Proceedings of the American Mathematical Society
1635:Gordan and Benson conjectured that if a compact
69:, sharing a certain cohomological property with
1757:"Examples of compact Lefschetz solvmanifolds"
8:
301:)-dimensional smooth manifold. Each element
112:
94:
73:, that of satisfying the conclusion of the
1772:
1731:
1599:
1579:
1559:
1514:
1478:
1452:
1416:
1389:
1357:
1320:
1288:
1249:
1241:
1213:
1205:
1192:
1181:
1175:
1138:
1111:
1103:
1081:
1073:
1060:
1049:
1043:
1007:
999:
977:
969:
950:
939:
933:
879:
871:
865:
829:
821:
815:
783:
741:
733:
705:
697:
684:
673:
667:
632:
605:
599:
579:
558:
547:
541:
515:
445:
420:
401:
395:
372:
335:
327:
309:
283:
263:
219:
218:
197:
183:
182:
161:
145:
133:
86:
41:
32:
1652:
7:
118:{\displaystyle k\in \{1,\ldots ,n\}}
14:
1543:The real manifold underlying any
1539:Where to find Lefschetz manifolds
353:{\displaystyle \in H_{DR}^{2}(M)}
62:{\displaystyle (M^{2n},\omega )}
1535:is a strong Lefschetz element.
903:{\displaystyle H_{DR}^{n+i}(M)}
853:{\displaystyle H_{DR}^{n-i}(M)}
1547:is a symplectic manifold. The
1522:
1516:
1492:
1480:
1460:
1454:
1430:
1418:
1371:
1359:
1328:
1322:
1267:
1261:
1234:
1231:
1225:
1188:
1182:
1146:
1140:
1096:
1093:
1087:
1056:
1050:
992:
989:
983:
946:
940:
897:
891:
847:
841:
759:
753:
726:
723:
717:
680:
674:
612:
606:
554:
548:
523:
517:
490:
478:
475:
472:
466:
460:
454:
438:
435:
429:
408:
402:
347:
341:
317:
311:
223:
209:
190:
187:
173:
151:
138:
56:
34:
1:
1574:{strong Lefschetz manifolds}
1308:{\displaystyle 0\leq i\leq n}
652:{\displaystyle 0\leq i\leq n}
1761:Tokyo Journal of Mathematics
1733:10.1016/0040-9383(88)90029-8
1473:is a Lefschetz element, and
1498:{\displaystyle (M,\omega )}
1436:{\displaystyle (M,\omega )}
1377:{\displaystyle (M,\omega )}
1819:
1283:is an isomorphism for all
1621:proved in 1988 that if a
1507:strong Lefschetz manifold
79:strong Lefschetz property
1607:{\displaystyle \subset }
1587:{\displaystyle \subset }
1567:{\displaystyle \subset }
1549:strong Lefschetz theorem
1343:strong Lefschetz element
567:{\displaystyle L_{}^{i}}
24:is a particular kind of
1755:Yamada, Takumi (2002).
1133:are isomorphisms, then
1774:10.3836/tjm/1244208853
1614:{symplectic manifolds}
1608:
1594:{Lefschetz manifolds}
1588:
1568:
1529:
1499:
1467:
1437:
1401:
1378:
1347:strong Lefschetz class
1335:
1309:
1274:
1153:
1124:
1026:
915:Hard Lefschetz theorem
904:
854:
792:
769:
653:
621:
588:
568:
530:
497:
381:
354:
295:
272:
230:
119:
77:. More precisely, the
75:Hard Lefschetz theorem
63:
1637:complete solvmanifold
1609:
1589:
1569:
1530:
1500:
1468:
1438:
1402:
1379:
1336:
1310:
1275:
1154:
1125:
1027:
905:
855:
793:
770:
654:
622:
589:
569:
531:
498:
382:
355:
296:
273:
231:
120:
64:
1661:Thurston, William P.
1598:
1578:
1558:
1513:
1477:
1451:
1415:
1388:
1356:
1319:
1287:
1174:
1137:
1042:
932:
864:
814:
782:
666:
631:
620:{\displaystyle L_{}}
598:
578:
540:
514:
394:
371:
308:
282:
262:
239:be an isomorphism.
132:
85:
31:
1803:Symplectic geometry
1554:{Kähler manifolds}
1409:symplectic manifold
1260:
1224:
1197:
1119:
1086:
1065:
1021:
982:
961:
890:
840:
752:
716:
689:
627:, we have for each
563:
340:
125:, the cup product
26:symplectic manifold
1710:Gordon, Carolyn S.
1604:
1584:
1564:
1525:
1495:
1463:
1445:Lefschetz manifold
1433:
1400:{\displaystyle 2n}
1397:
1374:
1331:
1305:
1270:
1237:
1201:
1177:
1149:
1120:
1099:
1069:
1045:
1022:
995:
965:
935:
900:
867:
850:
817:
788:
765:
729:
693:
669:
649:
617:
584:
564:
543:
526:
493:
377:
365:de Rham cohomology
350:
323:
294:{\displaystyle 2n}
291:
268:
226:
115:
59:
22:Lefschetz manifold
1619:Carolyn S. Gordon
1161:Lefschetz element
791:{\displaystyle M}
587:{\displaystyle i}
380:{\displaystyle M}
271:{\displaystyle M}
1810:
1787:
1786:
1776:
1752:
1746:
1745:
1735:
1705:
1699:
1698:
1657:
1617:Chal Benson and
1613:
1611:
1610:
1605:
1593:
1591:
1590:
1585:
1573:
1571:
1570:
1565:
1534:
1532:
1531:
1528:{\displaystyle }
1526:
1504:
1502:
1501:
1496:
1472:
1470:
1469:
1466:{\displaystyle }
1464:
1442:
1440:
1439:
1434:
1406:
1404:
1403:
1398:
1383:
1381:
1380:
1375:
1340:
1338:
1337:
1334:{\displaystyle }
1332:
1314:
1312:
1311:
1306:
1279:
1277:
1276:
1271:
1259:
1248:
1223:
1212:
1196:
1191:
1158:
1156:
1155:
1152:{\displaystyle }
1150:
1129:
1127:
1126:
1121:
1118:
1110:
1085:
1080:
1064:
1059:
1031:
1029:
1028:
1023:
1020:
1006:
981:
976:
960:
949:
909:
907:
906:
901:
889:
878:
859:
857:
856:
851:
839:
828:
808:Poincaré duality
797:
795:
794:
789:
774:
772:
771:
766:
751:
740:
715:
704:
688:
683:
658:
656:
655:
650:
626:
624:
623:
618:
616:
615:
594:th iteration of
593:
591:
590:
585:
573:
571:
570:
565:
562:
557:
535:
533:
532:
529:{\displaystyle }
527:
502:
500:
499:
494:
453:
452:
428:
427:
412:
411:
386:
384:
383:
378:
359:
357:
356:
351:
339:
334:
300:
298:
297:
292:
277:
275:
274:
269:
248:William Thurston
235:
233:
232:
227:
222:
208:
207:
186:
172:
171:
150:
149:
124:
122:
121:
116:
71:Kähler manifolds
68:
66:
65:
60:
49:
48:
1818:
1817:
1813:
1812:
1811:
1809:
1808:
1807:
1793:
1792:
1791:
1790:
1754:
1753:
1749:
1707:
1706:
1702:
1679:10.2307/2041749
1659:
1658:
1654:
1649:
1615:
1596:
1595:
1576:
1575:
1556:
1555:
1545:Kähler manifold
1541:
1511:
1510:
1475:
1474:
1449:
1448:
1413:
1412:
1386:
1385:
1354:
1353:
1317:
1316:
1285:
1284:
1172:
1171:
1165:Lefschetz class
1135:
1134:
1040:
1039:
930:
929:
923:
862:
861:
812:
811:
780:
779:
664:
663:
629:
628:
601:
596:
595:
576:
575:
538:
537:
512:
511:
441:
416:
397:
392:
391:
387:induces a map
369:
368:
306:
305:
280:
279:
260:
259:
256:
193:
157:
141:
130:
129:
83:
82:
37:
29:
28:
12:
11:
5:
1816:
1814:
1806:
1805:
1795:
1794:
1789:
1788:
1767:(2): 261–283.
1747:
1726:(4): 513–518.
1708:Benson, Chal;
1700:
1651:
1650:
1648:
1645:
1603:
1583:
1563:
1553:
1540:
1537:
1524:
1521:
1518:
1494:
1491:
1488:
1485:
1482:
1462:
1459:
1456:
1432:
1429:
1426:
1423:
1420:
1396:
1393:
1373:
1370:
1367:
1364:
1361:
1330:
1327:
1324:
1304:
1301:
1298:
1295:
1292:
1281:
1280:
1269:
1266:
1263:
1258:
1255:
1252:
1247:
1244:
1240:
1236:
1233:
1230:
1227:
1222:
1219:
1216:
1211:
1208:
1204:
1200:
1195:
1190:
1187:
1184:
1180:
1148:
1145:
1142:
1131:
1130:
1117:
1114:
1109:
1106:
1102:
1098:
1095:
1092:
1089:
1084:
1079:
1076:
1072:
1068:
1063:
1058:
1055:
1052:
1048:
1033:
1032:
1019:
1016:
1013:
1010:
1005:
1002:
998:
994:
991:
988:
985:
980:
975:
972:
968:
964:
959:
956:
953:
948:
945:
942:
938:
922:
919:
899:
896:
893:
888:
885:
882:
877:
874:
870:
849:
846:
843:
838:
835:
832:
827:
824:
820:
810:tells us that
787:
776:
775:
764:
761:
758:
755:
750:
747:
744:
739:
736:
732:
728:
725:
722:
719:
714:
711:
708:
703:
700:
696:
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687:
682:
679:
676:
672:
648:
645:
642:
639:
636:
614:
611:
608:
604:
583:
561:
556:
553:
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546:
525:
522:
519:
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471:
468:
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459:
456:
451:
448:
444:
440:
437:
434:
431:
426:
423:
419:
415:
410:
407:
404:
400:
376:
363:of the second
361:
360:
349:
346:
343:
338:
333:
330:
326:
322:
319:
316:
313:
290:
287:
267:
255:
254:Lefschetz maps
252:
237:
236:
225:
221:
217:
214:
211:
206:
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200:
196:
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185:
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156:
153:
148:
144:
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137:
114:
111:
108:
105:
102:
99:
96:
93:
90:
81:asks that for
58:
55:
52:
47:
44:
40:
36:
13:
10:
9:
6:
4:
3:
2:
1815:
1804:
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1784:
1780:
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1770:
1766:
1762:
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1751:
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1739:
1734:
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1725:
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1711:
1704:
1701:
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1662:
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1620:
1601:
1581:
1561:
1552:
1550:
1546:
1538:
1536:
1519:
1508:
1489:
1486:
1483:
1457:
1446:
1427:
1424:
1421:
1410:
1407:-dimensional
1394:
1391:
1368:
1365:
1362:
1350:
1348:
1344:
1325:
1302:
1299:
1296:
1293:
1290:
1264:
1256:
1253:
1250:
1245:
1242:
1238:
1228:
1220:
1217:
1214:
1209:
1206:
1202:
1198:
1193:
1185:
1178:
1170:
1169:
1168:
1166:
1162:
1143:
1115:
1112:
1107:
1104:
1100:
1090:
1082:
1077:
1074:
1070:
1066:
1061:
1053:
1046:
1038:
1037:
1036:
1017:
1014:
1011:
1008:
1003:
1000:
996:
986:
978:
973:
970:
966:
962:
957:
954:
951:
943:
936:
928:
927:
926:
920:
918:
916:
911:
894:
886:
883:
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875:
872:
868:
844:
836:
833:
830:
825:
822:
818:
809:
805:
801:
785:
762:
756:
748:
745:
742:
737:
734:
730:
720:
712:
709:
706:
701:
698:
694:
690:
685:
677:
670:
662:
661:
660:
646:
643:
640:
637:
634:
609:
602:
581:
559:
551:
544:
520:
509:
508:Lefschetz map
487:
484:
481:
469:
463:
457:
449:
446:
442:
432:
424:
421:
417:
413:
405:
398:
390:
389:
388:
374:
366:
344:
336:
331:
328:
324:
320:
314:
304:
303:
302:
288:
285:
265:
253:
251:
249:
245:
244:Betti numbers
240:
215:
212:
204:
201:
198:
194:
179:
176:
168:
165:
162:
158:
154:
146:
142:
135:
128:
127:
126:
109:
106:
103:
100:
97:
91:
88:
80:
76:
72:
53:
50:
45:
42:
38:
27:
23:
19:
1764:
1760:
1750:
1723:
1717:
1703:
1670:
1664:
1655:
1634:
1616:
1542:
1506:
1444:
1351:
1346:
1342:
1282:
1164:
1160:
1132:
1034:
924:
912:
777:
507:
505:
362:
257:
241:
238:
78:
21:
15:
1626:nilmanifold
921:Definitions
506:called the
18:mathematics
1673:(2): 467.
536:. Letting
1602:⊂
1582:⊂
1562:⊂
1520:ω
1490:ω
1458:ω
1428:ω
1369:ω
1326:ω
1300:≤
1294:≤
1235:→
1218:−
1186:ω
1144:ω
1097:→
1054:ω
1015:−
993:→
955:−
944:ω
834:−
727:→
710:−
678:ω
644:≤
638:≤
610:ω
552:ω
521:ω
488:α
485:∧
482:ω
476:↦
470:α
439:→
406:ω
367:space of
321:∈
315:ω
191:→
166:−
155::
143:ω
136:∪
104:…
92:∈
54:ω
1797:Category
1719:Topology
1712:(1988).
804:oriented
1783:1948664
1742:0976592
1695:0402764
1687:2041749
1623:compact
1345:, or a
1315:, then
806:, then
800:compact
659:a map
574:be the
1781:
1740:
1693:
1685:
1167:. If
278:be a (
1683:JSTOR
1647:Notes
1641:torus
1630:torus
1505:is a
1443:is a
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1163:, or
1159:is a
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925:If
913:The
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258:Let
20:, a
1769:doi
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778:If
510:of
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