1137:, therefore, can be understood if one understands how hyperplane sections are identified across the slits and at the singular points. Away from the singular points, the identification can be described inductively. At the singular points, the
605:
464:
323:
1966:
1889:
1828:
2258:
173:
86:
for homology groups of complex algebraic varieties. Similar results have since been found for homotopy groups, in positive characteristic, and in other homology and cohomology theories.
1975:
nearly yields the usual
Lefschetz theorem for cohomology with coefficients in any field of characteristic zero. It is, however, slightly weaker because of the additional assumptions on
1691:
2102:
816:
731:
1621:
2036:
245:
893:
2996:
2390:
2348:
1729:
925:
848:
763:
637:
496:
355:
1767:
1561:
1021:
2128:
2439:
2413:, with the isomorphism in de Rham cohomology given by multiplication by a power of the class of the Kähler form. It can fail for non-Kähler manifolds: for example,
2156:
1347:
Neither
Lefschetz's proof nor Andreotti and Frankel's proof directly imply the Lefschetz hyperplane theorem for homotopy groups. An approach that does was found by
1095:
1070:-plane and making an additional finite number of slits, the resulting family of hyperplane sections is topologically trivial. That is, it is a product of a generic
1048:
988:
669:
528:
387:
1249:
2302:
2282:
2224:
2204:
2056:
1993:
1581:
1513:
1493:
1473:
1453:
1433:
1413:
1393:
1373:
1337:
1317:
1297:
1273:
1226:
1198:
1159:
1135:
1115:
1068:
961:
213:
193:
139:
119:
2748:
2158:-adic cohomology. Up to some restrictions on the constructible sheaf, the Lefschetz theorem remains true for constructible sheaves in positive characteristic.
2138:
The motivation behind Artin and
Grothendieck's proof for constructible sheaves was to give a proof that could be adapted to the setting of Ă©tale and
2698:
2010:
found a generalization of the
Lefschetz hyperplane theorem to the case where the coefficients of the cohomology lie not in a field but instead in a
679:, one can show that each of these statements is equivalent to a vanishing theorem for certain relative topological invariants. In order, these are:
3011:
3006:
536:
2970:
2882:
2845:
2812:
1892:
395:
254:
2828:, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics , vol. 48, Berlin, New York:
2926:
1898:
1833:
1772:
1201:
2703:
1972:
2417:
have vanishing second cohomology groups, so there is no analogue of the second cohomology class of a hyperplane section.
2954:
2907:
2229:
144:
1626:
2869:, Collection de Monographies publiée sous la Direction de M. Émile Borel (in French), Paris: Gauthier-Villars
2061:
1530:
found that under certain restrictions, it is possible to prove a
Lefschetz-type theorem for the Hodge groups
771:
686:
1586:
2681:
2007:
63:
3001:
2629:
2162:
90:
2935:
2017:
218:
2800:
2686:
2421:
2011:
856:
676:
2803:(1992), "Solomon Lefschetz", in National Academy of Sciences, Office of the Home Secretary (ed.),
2781:
2654:
2181:
1161:
of a particularly simple form. This coordinate system can be used to prove the theorem directly.
56:
36:
32:
2409:
2353:
2311:
1696:
898:
821:
736:
2966:
2878:
2862:
2841:
2821:
2808:
2796:
2792:
2765:
2646:
2443:
of smooth projective varieties over algebraically closed fields of positive characteristic by
1527:
1276:
936:
610:
469:
328:
83:
44:
1734:
1533:
993:
2958:
2833:
2757:
2712:
2673:
2638:
2624:
2305:
2107:
1523:
1395:
of a section of a line bundle. An application of Morse theory to this section implies that
1173:
940:
52:
2980:
2915:
2892:
2855:
2777:
2726:
2666:
2424:
2141:
1073:
1026:
966:
642:
501:
360:
2976:
2911:
2888:
2851:
2829:
2773:
2722:
2662:
2261:
1231:
17:
2743:
2739:
2620:
2444:
2287:
2267:
2209:
2189:
2041:
1978:
1566:
1498:
1478:
1458:
1438:
1418:
1398:
1378:
1358:
1322:
1302:
1282:
1258:
1211:
1205:
1183:
1169:
1144:
1120:
1100:
1053:
946:
198:
178:
124:
104:
71:
47:
and the shape of its subvarieties. More precisely, the theorem says that for a variety
2990:
2946:
2922:
2785:
2003:
2403:. It immediately implies the injectivity part of the Lefschetz hyperplane theorem.
1769:. By Hodge theory, these cohomology groups are equal to the sheaf cohomology groups
2414:
2169:
1177:
1348:
1455:
or more. From this, it follows that the relative homology and homotopy groups of
2899:
1138:
28:
2837:
2694:
1352:
1252:
67:
2962:
2769:
2717:
2650:
1023:. Because a generic hyperplane section is smooth, all but a finite number of
600:{\displaystyle \pi _{k}(Y,\mathbb {Z} )\rightarrow \pi _{k}(X,\mathbb {Z} )}
247:
is smooth. The
Lefschetz theorem refers to any of the following statements:
89:
A far-reaching generalization of the hard
Lefschetz theorem is given by the
1200:
plays the role of a Morse function. The basic tool in this approach is the
2761:
2658:
1339:. The long exact sequence of relative homology then gives the theorem.
459:{\displaystyle H^{k}(X,\mathbb {Z} )\rightarrow H^{k}(Y,\mathbb {Z} )}
318:{\displaystyle H_{k}(Y,\mathbb {Z} )\rightarrow H_{k}(X,\mathbb {Z} )}
943:
to prove the theorem. Rather than considering the hyperplane section
2642:
43:
is a precise statement of certain relations between the shape of an
1961:{\displaystyle H^{q}(X,\textstyle \bigwedge ^{p}\Omega _{X}|_{Y})}
2928:
Théorie de Hodge et théorème de
Lefschetz « difficile »
2165:. In this setting, the theorem holds for highly singular spaces.
97:
The
Lefschetz hyperplane theorem for complex projective varieties
2399:, christened in French by Grothendieck more colloquially as the
1884:{\displaystyle H^{q}(Y,\textstyle \bigwedge ^{p}\Omega _{Y})}
1823:{\displaystyle H^{q}(X,\textstyle \bigwedge ^{p}\Omega _{X})}
2086:
2023:
1593:
2953:, Cambridge Studies in Advanced Mathematics, vol. 77,
1050:
are smooth varieties. After removing these points from the
1176:
recognized that
Lefschetz's theorem could be recast using
2501:
2627:(1959), "The Lefschetz theorem on hyperplane sections",
2226:-dimensional non-singular complex projective variety in
1999:
Artin and Grothendieck's proof for constructible sheaves
1141:
implies that there is a choice of coordinate system for
1351:
no later than 1957 and was simplified and published by
1921:
1856:
1795:
963:
alone, he put it into a family of hyperplane sections
2427:
2356:
2314:
2290:
2270:
2232:
2212:
2192:
2144:
2110:
2064:
2044:
2020:
1981:
1901:
1836:
1775:
1737:
1699:
1629:
1589:
1569:
1536:
1501:
1481:
1461:
1441:
1421:
1401:
1381:
1361:
1325:
1305:
1285:
1261:
1234:
1214:
1186:
1147:
1123:
1103:
1076:
1056:
1029:
996:
969:
949:
901:
859:
824:
774:
739:
689:
645:
613:
539:
504:
472:
398:
363:
331:
257:
221:
201:
181:
147:
141:-dimensional complex projective algebraic variety in
127:
107:
1891:. Therefore, the theorem follows from applying the
2433:
2384:
2342:
2296:
2276:
2252:
2218:
2198:
2150:
2134:The Lefschetz theorem in other cohomology theories
2122:
2096:
2050:
2030:
1987:
1960:
1883:
1822:
1761:
1723:
1685:
1615:
1575:
1555:
1507:
1487:
1467:
1447:
1427:
1407:
1387:
1367:
1331:
1311:
1291:
1267:
1243:
1220:
1192:
1153:
1129:
1109:
1089:
1062:
1042:
1015:
982:
955:
919:
887:
842:
810:
757:
725:
663:
631:
599:
522:
490:
458:
381:
349:
317:
239:
207:
187:
167:
133:
113:
2512:
2951:Hodge theory and complex algebraic geometry. II
1998:
2807:, vol. 61, The National Academies Press,
2406:The hard Lefschetz theorem in fact holds for
2308:of a hyperplane gives an isomorphism between
466:in singular cohomology is an isomorphism for
8:
2253:{\displaystyle \mathbb {C} \mathbf {P} ^{N}}
2014:. They prove that for a constructible sheaf
1519:Kodaira and Spencer's proof for Hodge groups
168:{\displaystyle \mathbb {C} \mathbf {P} ^{N}}
82:. A result of this kind was first stated by
2867:L'Analysis situs et la géométrie algébrique
325:in singular homology is an isomorphism for
2582:
2570:
2546:
2420:The hard Lefschetz theorem was proven for
2997:Topological methods of algebraic geometry
2906:, Annals of Mathematics Studies, No. 51,
2716:
2685:
2490:
2426:
2361:
2355:
2319:
2313:
2289:
2269:
2244:
2239:
2234:
2233:
2231:
2211:
2191:
2143:
2109:
2085:
2084:
2069:
2063:
2043:
2022:
2021:
2019:
1980:
1948:
1943:
1936:
1926:
1906:
1900:
1871:
1861:
1841:
1835:
1810:
1800:
1780:
1774:
1736:
1698:
1662:
1634:
1628:
1598:
1592:
1591:
1588:
1568:
1541:
1535:
1500:
1480:
1460:
1440:
1420:
1400:
1380:
1360:
1324:
1304:
1284:
1260:
1233:
1213:
1185:
1146:
1122:
1102:
1081:
1075:
1055:
1034:
1028:
1007:
995:
974:
968:
948:
900:
864:
858:
823:
801:
800:
779:
773:
738:
716:
715:
694:
688:
644:
612:
590:
589:
574:
560:
559:
544:
538:
503:
471:
449:
448:
433:
419:
418:
403:
397:
362:
330:
308:
307:
292:
278:
277:
262:
256:
220:
200:
180:
159:
154:
149:
148:
146:
126:
106:
2168:A Lefschetz-type theorem also holds for
1686:{\displaystyle H^{p,q}(X)\to H^{p,q}(Y)}
768:The relative singular cohomology groups
2502:Griffiths, Spencer & Whitehead 1992
2459:
2448:
2161:The theorem can also be generalized to
2097:{\displaystyle H^{k}(U,{\mathcal {F}})}
811:{\displaystyle H^{k}(X,Y;\mathbb {Z} )}
726:{\displaystyle H_{k}(X,Y;\mathbb {Z} )}
231:
2605:
2558:
2523:
2478:
2466:
1518:
1515:and higher, which yields the theorem.
683:The relative singular homology groups
2594:
1616:{\displaystyle {\mathcal {O}}_{X}(Y)}
7:
2877:, New York: Chelsea Publishing Co.,
2749:Publications Mathématiques de l'IHÉS
2535:
2826:Positivity in algebraic geometry. I
1623:is ample. Then the restriction map
1583:is smooth and that the line bundle
1933:
1868:
1807:
25:
1968:and using a long exact sequence.
1355:in 1959. Thom and Bott interpret
1342:
2240:
1893:Akizuki–Nakano vanishing theorem
1435:by adjoining cells of dimension
1319:are trivial in degree less than
155:
2469:, Theorem 7.3 and Corollary 7.4
3012:Theorems in algebraic topology
3007:Theorems in algebraic geometry
2379:
2373:
2337:
2331:
2091:
2075:
2031:{\displaystyle {\mathcal {F}}}
1971:Combining this proof with the
1954:
1944:
1912:
1877:
1847:
1816:
1786:
1680:
1674:
1655:
1652:
1646:
1610:
1604:
1204:, which states that a complex
882:
870:
805:
785:
720:
700:
594:
580:
567:
564:
550:
453:
439:
426:
423:
409:
312:
298:
285:
282:
268:
240:{\displaystyle U=X\setminus Y}
1:
2704:Michigan Mathematical Journal
1973:universal coefficient theorem
1251:) has the homotopy type of a
1165:Andreotti and Frankel's proof
888:{\displaystyle \pi _{k}(X,Y)}
853:The relative homotopy groups
2513:Andreotti & Frankel 1959
1563:. Specifically, assume that
1495:are concentrated in degrees
41:Lefschetz hyperplane theorem
2873:Lefschetz, Solomon (1971),
2744:"La conjecture de Weil. II"
2699:"On a theorem of Lefschetz"
2401:Théorème de Lefschetz vache
1097:with an open subset of the
195:be a hyperplane section of
3028:
2955:Cambridge University Press
2908:Princeton University Press
2385:{\displaystyle H^{n+k}(X)}
2343:{\displaystyle H^{n-k}(X)}
2179:
1724:{\displaystyle p+q<n-1}
1375:as the vanishing locus in
2838:10.1007/978-3-642-18808-4
1228:(and thus real dimension
1202:Andreotti–Frankel theorem
920:{\displaystyle k\leq n-1}
843:{\displaystyle k\leq n-1}
758:{\displaystyle k\leq n-1}
2963:10.1017/CBO9780511615177
2058:, the cohomology groups
1415:can be constructed from
1343:Thom's and Bott's proofs
1275:. This implies that the
632:{\displaystyle k<n-1}
491:{\displaystyle k<n-1}
350:{\displaystyle k<n-1}
18:Strong Lefschetz theorem
2304:-fold product with the
1762:{\displaystyle p+q=n-1}
1556:{\displaystyle H^{p,q}}
1016:{\displaystyle Y=Y_{0}}
2718:10.1307/mmj/1028998225
2435:
2397:hard Lefschetz theorem
2386:
2344:
2298:
2278:
2254:
2220:
2200:
2176:Hard Lefschetz theorem
2152:
2124:
2123:{\displaystyle k>n}
2098:
2052:
2032:
2008:Alexander Grothendieck
1989:
1962:
1885:
1824:
1763:
1725:
1687:
1617:
1577:
1557:
1509:
1489:
1469:
1449:
1429:
1409:
1389:
1369:
1333:
1313:
1293:
1269:
1245:
1222:
1194:
1155:
1131:
1111:
1091:
1064:
1044:
1017:
984:
957:
921:
889:
844:
812:
759:
727:
665:
639:and is surjective for
633:
607:is an isomorphism for
601:
524:
492:
460:
383:
357:and is surjective for
351:
319:
241:
209:
189:
169:
135:
115:
2630:Annals of Mathematics
2436:
2434:{\displaystyle \ell }
2387:
2345:
2299:
2279:
2255:
2221:
2201:
2163:intersection homology
2153:
2151:{\displaystyle \ell }
2125:
2099:
2053:
2038:on an affine variety
2033:
1990:
1963:
1886:
1825:
1764:
1726:
1693:is an isomorphism if
1688:
1618:
1578:
1558:
1510:
1490:
1470:
1450:
1430:
1410:
1390:
1370:
1334:
1314:
1294:
1270:
1246:
1223:
1208:of complex dimension
1195:
1180:. Here the parameter
1164:
1156:
1132:
1112:
1092:
1090:{\displaystyle Y_{t}}
1065:
1045:
1043:{\displaystyle Y_{t}}
1018:
985:
983:{\displaystyle Y_{t}}
958:
931:
922:
890:
845:
813:
760:
728:
666:
664:{\displaystyle k=n-1}
634:
602:
525:
523:{\displaystyle k=n-1}
498:and is injective for
493:
461:
384:
382:{\displaystyle k=n-1}
352:
320:
242:
210:
190:
170:
136:
116:
91:decomposition theorem
2900:Milnor, John Willard
2805:Biographical Memoirs
2801:Whitehead, George W.
2678:The Hodge Conjecture
2425:
2354:
2312:
2288:
2268:
2230:
2210:
2190:
2142:
2108:
2062:
2042:
2018:
1979:
1899:
1834:
1773:
1735:
1731:and is injective if
1697:
1627:
1587:
1567:
1534:
1499:
1479:
1459:
1439:
1419:
1399:
1379:
1359:
1323:
1303:
1283:
1259:
1255:of (real) dimension
1232:
1212:
1184:
1145:
1121:
1101:
1074:
1054:
1027:
994:
967:
947:
899:
857:
822:
772:
737:
687:
643:
611:
537:
502:
470:
396:
361:
329:
255:
219:
199:
179:
145:
125:
105:
2012:constructible sheaf
939:used his idea of a
677:long exact sequence
78:determine those of
2863:Lefschetz, Solomon
2822:Lazarsfeld, Robert
2797:Spencer, Donald C.
2793:Griffiths, Phillip
2762:10.1007/BF02684780
2445:Pierre Deligne
2431:
2382:
2340:
2294:
2274:
2250:
2216:
2196:
2182:Lefschetz manifold
2148:
2120:
2094:
2048:
2028:
1985:
1958:
1957:
1881:
1880:
1820:
1819:
1759:
1721:
1683:
1613:
1573:
1553:
1505:
1485:
1465:
1445:
1425:
1405:
1385:
1365:
1329:
1309:
1289:
1265:
1244:{\displaystyle 2n}
1241:
1218:
1190:
1151:
1127:
1107:
1087:
1060:
1040:
1013:
980:
953:
917:
885:
840:
808:
755:
723:
661:
629:
597:
520:
488:
456:
379:
347:
315:
237:
205:
185:
165:
131:
111:
57:hyperplane section
37:algebraic topology
33:algebraic geometry
31:, specifically in
2972:978-0-521-80283-3
2884:978-0-8284-0234-7
2847:978-3-540-22533-1
2814:978-0-309-04746-3
2674:Beauville, Arnaud
2633:, Second Series,
2625:Frankel, Theodore
2297:{\displaystyle k}
2277:{\displaystyle X}
2219:{\displaystyle n}
2199:{\displaystyle X}
2051:{\displaystyle U}
1988:{\displaystyle Y}
1931:
1866:
1805:
1576:{\displaystyle Y}
1528:Donald C. Spencer
1508:{\displaystyle n}
1488:{\displaystyle X}
1468:{\displaystyle Y}
1448:{\displaystyle n}
1428:{\displaystyle Y}
1408:{\displaystyle X}
1388:{\displaystyle X}
1368:{\displaystyle Y}
1332:{\displaystyle n}
1312:{\displaystyle X}
1292:{\displaystyle Y}
1277:relative homology
1268:{\displaystyle n}
1221:{\displaystyle n}
1193:{\displaystyle t}
1154:{\displaystyle X}
1130:{\displaystyle X}
1110:{\displaystyle t}
1063:{\displaystyle t}
956:{\displaystyle Y}
937:Solomon Lefschetz
932:Lefschetz's proof
208:{\displaystyle X}
188:{\displaystyle Y}
134:{\displaystyle n}
114:{\displaystyle X}
84:Solomon Lefschetz
45:algebraic variety
16:(Redirected from
3019:
2983:
2942:
2940:
2934:, archived from
2933:
2918:
2895:
2870:
2858:
2817:
2788:
2735:
2734:
2733:
2720:
2690:
2689:
2669:
2608:
2603:
2597:
2592:
2586:
2585:, Example 3.1.25
2580:
2574:
2573:, Theorem 3.1.13
2568:
2562:
2556:
2550:
2549:, Example 3.1.24
2544:
2538:
2533:
2527:
2521:
2515:
2510:
2504:
2499:
2493:
2488:
2482:
2476:
2470:
2464:
2441:-adic cohomology
2440:
2438:
2437:
2432:
2391:
2389:
2388:
2383:
2372:
2371:
2349:
2347:
2346:
2341:
2330:
2329:
2306:cohomology class
2303:
2301:
2300:
2295:
2283:
2281:
2280:
2275:
2259:
2257:
2256:
2251:
2249:
2248:
2243:
2237:
2225:
2223:
2222:
2217:
2205:
2203:
2202:
2197:
2157:
2155:
2154:
2149:
2129:
2127:
2126:
2121:
2104:vanish whenever
2103:
2101:
2100:
2095:
2090:
2089:
2074:
2073:
2057:
2055:
2054:
2049:
2037:
2035:
2034:
2029:
2027:
2026:
1994:
1992:
1991:
1986:
1967:
1965:
1964:
1959:
1953:
1952:
1947:
1941:
1940:
1930:
1922:
1911:
1910:
1890:
1888:
1887:
1882:
1876:
1875:
1865:
1857:
1846:
1845:
1829:
1827:
1826:
1821:
1815:
1814:
1804:
1796:
1785:
1784:
1768:
1766:
1765:
1760:
1730:
1728:
1727:
1722:
1692:
1690:
1689:
1684:
1673:
1672:
1645:
1644:
1622:
1620:
1619:
1614:
1603:
1602:
1597:
1596:
1582:
1580:
1579:
1574:
1562:
1560:
1559:
1554:
1552:
1551:
1524:Kunihiko Kodaira
1514:
1512:
1511:
1506:
1494:
1492:
1491:
1486:
1474:
1472:
1471:
1466:
1454:
1452:
1451:
1446:
1434:
1432:
1431:
1426:
1414:
1412:
1411:
1406:
1394:
1392:
1391:
1386:
1374:
1372:
1371:
1366:
1338:
1336:
1335:
1330:
1318:
1316:
1315:
1310:
1298:
1296:
1295:
1290:
1274:
1272:
1271:
1266:
1250:
1248:
1247:
1242:
1227:
1225:
1224:
1219:
1199:
1197:
1196:
1191:
1174:Theodore Frankel
1160:
1158:
1157:
1152:
1136:
1134:
1133:
1128:
1116:
1114:
1113:
1108:
1096:
1094:
1093:
1088:
1086:
1085:
1069:
1067:
1066:
1061:
1049:
1047:
1046:
1041:
1039:
1038:
1022:
1020:
1019:
1014:
1012:
1011:
989:
987:
986:
981:
979:
978:
962:
960:
959:
954:
941:Lefschetz pencil
926:
924:
923:
918:
894:
892:
891:
886:
869:
868:
849:
847:
846:
841:
817:
815:
814:
809:
804:
784:
783:
764:
762:
761:
756:
732:
730:
729:
724:
719:
699:
698:
670:
668:
667:
662:
638:
636:
635:
630:
606:
604:
603:
598:
593:
579:
578:
563:
549:
548:
533:The natural map
529:
527:
526:
521:
497:
495:
494:
489:
465:
463:
462:
457:
452:
438:
437:
422:
408:
407:
392:The natural map
388:
386:
385:
380:
356:
354:
353:
348:
324:
322:
321:
316:
311:
297:
296:
281:
267:
266:
251:The natural map
246:
244:
243:
238:
214:
212:
211:
206:
194:
192:
191:
186:
174:
172:
171:
166:
164:
163:
158:
152:
140:
138:
137:
132:
120:
118:
117:
112:
53:projective space
21:
3027:
3026:
3022:
3021:
3020:
3018:
3017:
3016:
2987:
2986:
2973:
2945:
2938:
2931:
2921:
2898:
2885:
2875:Selected papers
2872:
2861:
2848:
2830:Springer-Verlag
2820:
2815:
2791:
2756:(52): 137–252,
2740:Deligne, Pierre
2738:
2731:
2729:
2693:
2672:
2643:10.2307/1970034
2621:Andreotti, Aldo
2619:
2616:
2611:
2604:
2600:
2593:
2589:
2583:Lazarsfeld 2004
2581:
2577:
2571:Lazarsfeld 2004
2569:
2565:
2557:
2553:
2547:Lazarsfeld 2004
2545:
2541:
2534:
2530:
2522:
2518:
2511:
2507:
2500:
2496:
2489:
2485:
2477:
2473:
2465:
2461:
2457:
2423:
2422:
2410:Kähler manifold
2357:
2352:
2351:
2315:
2310:
2309:
2286:
2285:
2266:
2265:
2262:cohomology ring
2238:
2228:
2227:
2208:
2207:
2188:
2187:
2184:
2178:
2140:
2139:
2136:
2106:
2105:
2065:
2060:
2059:
2040:
2039:
2016:
2015:
2001:
1977:
1976:
1942:
1932:
1902:
1897:
1896:
1867:
1837:
1832:
1831:
1806:
1776:
1771:
1770:
1733:
1732:
1695:
1694:
1658:
1630:
1625:
1624:
1590:
1585:
1584:
1565:
1564:
1537:
1532:
1531:
1521:
1497:
1496:
1477:
1476:
1457:
1456:
1437:
1436:
1417:
1416:
1397:
1396:
1377:
1376:
1357:
1356:
1345:
1321:
1320:
1301:
1300:
1281:
1280:
1257:
1256:
1230:
1229:
1210:
1209:
1182:
1181:
1167:
1143:
1142:
1119:
1118:
1099:
1098:
1077:
1072:
1071:
1052:
1051:
1030:
1025:
1024:
1003:
992:
991:
970:
965:
964:
945:
944:
934:
897:
896:
860:
855:
854:
820:
819:
775:
770:
769:
735:
734:
690:
685:
684:
641:
640:
609:
608:
570:
540:
535:
534:
500:
499:
468:
467:
429:
399:
394:
393:
359:
358:
327:
326:
288:
258:
253:
252:
217:
216:
197:
196:
177:
176:
153:
143:
142:
123:
122:
103:
102:
99:
72:homotopy groups
23:
22:
15:
12:
11:
5:
3025:
3023:
3015:
3014:
3009:
3004:
2999:
2989:
2988:
2985:
2984:
2971:
2947:Voisin, Claire
2943:
2923:Sabbah, Claude
2919:
2896:
2883:
2859:
2846:
2818:
2813:
2789:
2736:
2711:(3): 211–216,
2691:
2687:10.1.1.74.2423
2670:
2637:(3): 713–717,
2615:
2612:
2610:
2609:
2598:
2587:
2575:
2563:
2561:, Theorem 1.29
2551:
2539:
2528:
2516:
2505:
2494:
2491:Lefschetz 1924
2483:
2481:, Theorem 1.23
2471:
2458:
2456:
2453:
2430:
2381:
2378:
2375:
2370:
2367:
2364:
2360:
2339:
2336:
2333:
2328:
2325:
2322:
2318:
2293:
2273:
2260:. Then in the
2247:
2242:
2236:
2215:
2195:
2177:
2174:
2147:
2135:
2132:
2119:
2116:
2113:
2093:
2088:
2083:
2080:
2077:
2072:
2068:
2047:
2025:
2000:
1997:
1984:
1956:
1951:
1946:
1939:
1935:
1929:
1925:
1920:
1917:
1914:
1909:
1905:
1879:
1874:
1870:
1864:
1860:
1855:
1852:
1849:
1844:
1840:
1818:
1813:
1809:
1803:
1799:
1794:
1791:
1788:
1783:
1779:
1758:
1755:
1752:
1749:
1746:
1743:
1740:
1720:
1717:
1714:
1711:
1708:
1705:
1702:
1682:
1679:
1676:
1671:
1668:
1665:
1661:
1657:
1654:
1651:
1648:
1643:
1640:
1637:
1633:
1612:
1609:
1606:
1601:
1595:
1572:
1550:
1547:
1544:
1540:
1520:
1517:
1504:
1484:
1464:
1444:
1424:
1404:
1384:
1364:
1344:
1341:
1328:
1308:
1288:
1264:
1240:
1237:
1217:
1206:affine variety
1189:
1170:Aldo Andreotti
1166:
1163:
1150:
1126:
1106:
1084:
1080:
1059:
1037:
1033:
1010:
1006:
1002:
999:
977:
973:
952:
933:
930:
929:
928:
916:
913:
910:
907:
904:
884:
881:
878:
875:
872:
867:
863:
851:
839:
836:
833:
830:
827:
807:
803:
799:
796:
793:
790:
787:
782:
778:
766:
754:
751:
748:
745:
742:
722:
718:
714:
711:
708:
705:
702:
697:
693:
673:
672:
660:
657:
654:
651:
648:
628:
625:
622:
619:
616:
596:
592:
588:
585:
582:
577:
573:
569:
566:
562:
558:
555:
552:
547:
543:
531:
519:
516:
513:
510:
507:
487:
484:
481:
478:
475:
455:
451:
447:
444:
441:
436:
432:
428:
425:
421:
417:
414:
411:
406:
402:
390:
378:
375:
372:
369:
366:
346:
343:
340:
337:
334:
314:
310:
306:
303:
300:
295:
291:
287:
284:
280:
276:
273:
270:
265:
261:
236:
233:
230:
227:
224:
204:
184:
162:
157:
151:
130:
110:
98:
95:
24:
14:
13:
10:
9:
6:
4:
3:
2:
3024:
3013:
3010:
3008:
3005:
3003:
3000:
2998:
2995:
2994:
2992:
2982:
2978:
2974:
2968:
2964:
2960:
2956:
2952:
2948:
2944:
2941:on 2004-07-07
2937:
2930:
2929:
2924:
2920:
2917:
2913:
2909:
2905:
2901:
2897:
2894:
2890:
2886:
2880:
2876:
2871:Reprinted in
2868:
2864:
2860:
2857:
2853:
2849:
2843:
2839:
2835:
2831:
2827:
2823:
2819:
2816:
2810:
2806:
2802:
2798:
2794:
2790:
2787:
2783:
2779:
2775:
2771:
2767:
2763:
2759:
2755:
2751:
2750:
2745:
2741:
2737:
2728:
2724:
2719:
2714:
2710:
2706:
2705:
2700:
2696:
2692:
2688:
2683:
2679:
2675:
2671:
2668:
2664:
2660:
2656:
2652:
2648:
2644:
2640:
2636:
2632:
2631:
2626:
2622:
2618:
2617:
2613:
2607:
2602:
2599:
2596:
2591:
2588:
2584:
2579:
2576:
2572:
2567:
2564:
2560:
2555:
2552:
2548:
2543:
2540:
2537:
2532:
2529:
2525:
2520:
2517:
2514:
2509:
2506:
2503:
2498:
2495:
2492:
2487:
2484:
2480:
2475:
2472:
2468:
2463:
2460:
2454:
2452:
2450:
2446:
2442:
2428:
2418:
2416:
2415:Hopf surfaces
2412:
2411:
2404:
2402:
2398:
2393:
2376:
2368:
2365:
2362:
2358:
2334:
2326:
2323:
2320:
2316:
2307:
2291:
2271:
2263:
2245:
2213:
2193:
2183:
2175:
2173:
2171:
2170:Picard groups
2166:
2164:
2159:
2145:
2133:
2131:
2117:
2114:
2111:
2081:
2078:
2070:
2066:
2045:
2013:
2009:
2005:
2004:Michael Artin
1996:
1982:
1974:
1969:
1949:
1937:
1927:
1923:
1918:
1915:
1907:
1903:
1894:
1872:
1862:
1858:
1853:
1850:
1842:
1838:
1811:
1801:
1797:
1792:
1789:
1781:
1777:
1756:
1753:
1750:
1747:
1744:
1741:
1738:
1718:
1715:
1712:
1709:
1706:
1703:
1700:
1677:
1669:
1666:
1663:
1659:
1649:
1641:
1638:
1635:
1631:
1607:
1599:
1570:
1548:
1545:
1542:
1538:
1529:
1525:
1516:
1502:
1482:
1462:
1442:
1422:
1402:
1382:
1362:
1354:
1350:
1340:
1326:
1306:
1286:
1278:
1262:
1254:
1238:
1235:
1215:
1207:
1203:
1187:
1179:
1175:
1171:
1162:
1148:
1140:
1124:
1104:
1082:
1078:
1057:
1035:
1031:
1008:
1004:
1000:
997:
975:
971:
950:
942:
938:
914:
911:
908:
905:
902:
895:are zero for
879:
876:
873:
865:
861:
852:
837:
834:
831:
828:
825:
818:are zero for
797:
794:
791:
788:
780:
776:
767:
752:
749:
746:
743:
740:
733:are zero for
712:
709:
706:
703:
695:
691:
682:
681:
680:
678:
658:
655:
652:
649:
646:
626:
623:
620:
617:
614:
586:
583:
575:
571:
556:
553:
545:
541:
532:
517:
514:
511:
508:
505:
485:
482:
479:
476:
473:
445:
442:
434:
430:
415:
412:
404:
400:
391:
376:
373:
370:
367:
364:
344:
341:
338:
335:
332:
304:
301:
293:
289:
274:
271:
263:
259:
250:
249:
248:
234:
228:
225:
222:
202:
182:
160:
128:
108:
96:
94:
92:
87:
85:
81:
77:
73:
69:
65:
61:
58:
54:
50:
46:
42:
38:
34:
30:
19:
3002:Morse theory
2950:
2936:the original
2927:
2904:Morse theory
2903:
2874:
2866:
2825:
2804:
2753:
2747:
2730:, retrieved
2708:
2702:
2677:
2634:
2628:
2614:Bibliography
2601:
2590:
2578:
2566:
2554:
2542:
2531:
2526:, p. 39
2519:
2508:
2497:
2486:
2474:
2462:
2419:
2408:any compact
2407:
2405:
2400:
2396:
2395:This is the
2394:
2185:
2167:
2160:
2137:
2002:
1970:
1522:
1346:
1178:Morse theory
1168:
935:
674:
100:
88:
79:
75:
59:
51:embedded in
48:
40:
26:
2695:Bott, Raoul
2606:Sabbah 2001
2559:Voisin 2003
2524:Milnor 1963
2479:Voisin 2003
2467:Milnor 1963
1139:Morse lemma
29:mathematics
2991:Categories
2732:2010-01-30
2455:References
2180:See also:
1353:Raoul Bott
1279:groups of
1253:CW-complex
215:such that
175:, and let
68:cohomology
2786:189769469
2770:1618-1913
2682:CiteSeerX
2651:0003-486X
2595:Beauville
2536:Bott 1959
2429:ℓ
2324:−
2146:ℓ
1934:Ω
1924:⋀
1869:Ω
1859:⋀
1808:Ω
1798:⋀
1754:−
1716:−
1656:→
1349:René Thom
912:−
906:≤
862:π
835:−
829:≤
750:−
744:≤
656:−
624:−
572:π
568:→
542:π
515:−
483:−
427:→
374:−
342:−
286:→
232:∖
2949:(2003),
2925:(2001),
2902:(1963),
2865:(1924),
2824:(2004),
2742:(1980),
2697:(1959),
1117:-plane.
990:, where
675:Using a
64:homology
2981:1997577
2916:0163331
2893:0299447
2856:2095471
2778:0601520
2727:0215323
2667:0177422
2659:1970034
2447: (
2979:
2969:
2914:
2891:
2881:
2854:
2844:
2811:
2784:
2776:
2768:
2725:
2684:
2665:
2657:
2649:
2284:, the
121:be an
70:, and
62:, the
55:and a
39:, the
2939:(PDF)
2932:(PDF)
2782:S2CID
2655:JSTOR
2206:be a
2967:ISBN
2879:ISBN
2842:ISBN
2809:ISBN
2766:ISSN
2647:ISSN
2449:1980
2350:and
2186:Let
2115:>
2006:and
1830:and
1710:<
1526:and
1172:and
618:<
477:<
336:<
101:Let
35:and
2959:doi
2834:doi
2758:doi
2713:doi
2639:doi
2451:).
2264:of
1895:to
1475:in
1299:in
74:of
27:In
2993::
2977:MR
2975:,
2965:,
2957:,
2912:MR
2910:,
2889:MR
2887:,
2852:MR
2850:,
2840:,
2832:,
2799:;
2795:;
2780:,
2774:MR
2772:,
2764:,
2754:52
2752:,
2746:,
2723:MR
2721:,
2707:,
2701:,
2680:,
2676:,
2663:MR
2661:,
2653:,
2645:,
2635:69
2623:;
2392:.
2172:.
2130:.
1995:.
93:.
66:,
2961::
2836::
2760::
2715::
2709:6
2641::
2380:)
2377:X
2374:(
2369:k
2366:+
2363:n
2359:H
2338:)
2335:X
2332:(
2327:k
2321:n
2317:H
2292:k
2272:X
2246:N
2241:P
2235:C
2214:n
2194:X
2118:n
2112:k
2092:)
2087:F
2082:,
2079:U
2076:(
2071:k
2067:H
2046:U
2024:F
1983:Y
1955:)
1950:Y
1945:|
1938:X
1928:p
1919:,
1916:X
1913:(
1908:q
1904:H
1878:)
1873:Y
1863:p
1854:,
1851:Y
1848:(
1843:q
1839:H
1817:)
1812:X
1802:p
1793:,
1790:X
1787:(
1782:q
1778:H
1757:1
1751:n
1748:=
1745:q
1742:+
1739:p
1719:1
1713:n
1707:q
1704:+
1701:p
1681:)
1678:Y
1675:(
1670:q
1667:,
1664:p
1660:H
1653:)
1650:X
1647:(
1642:q
1639:,
1636:p
1632:H
1611:)
1608:Y
1605:(
1600:X
1594:O
1571:Y
1549:q
1546:,
1543:p
1539:H
1503:n
1483:X
1463:Y
1443:n
1423:Y
1403:X
1383:X
1363:Y
1327:n
1307:X
1287:Y
1263:n
1239:n
1236:2
1216:n
1188:t
1149:X
1125:X
1105:t
1083:t
1079:Y
1058:t
1036:t
1032:Y
1009:0
1005:Y
1001:=
998:Y
976:t
972:Y
951:Y
927:.
915:1
909:n
903:k
883:)
880:Y
877:,
874:X
871:(
866:k
850:.
838:1
832:n
826:k
806:)
802:Z
798:;
795:Y
792:,
789:X
786:(
781:k
777:H
765:.
753:1
747:n
741:k
721:)
717:Z
713:;
710:Y
707:,
704:X
701:(
696:k
692:H
671:.
659:1
653:n
650:=
647:k
627:1
621:n
615:k
595:)
591:Z
587:,
584:X
581:(
576:k
565:)
561:Z
557:,
554:Y
551:(
546:k
530:.
518:1
512:n
509:=
506:k
486:1
480:n
474:k
454:)
450:Z
446:,
443:Y
440:(
435:k
431:H
424:)
420:Z
416:,
413:X
410:(
405:k
401:H
389:.
377:1
371:n
368:=
365:k
345:1
339:n
333:k
313:)
309:Z
305:,
302:X
299:(
294:k
290:H
283:)
279:Z
275:,
272:Y
269:(
264:k
260:H
235:Y
229:X
226:=
223:U
203:X
183:Y
161:N
156:P
150:C
129:n
109:X
80:Y
76:X
60:Y
49:X
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.