508:
415:
266:
572:
80:
142:
showed that the inequality is best possible by finding infinitely many cases where equality holds. The inequality is false in positive characteristic: William E. Lang (
699:
660:
331:
581:
the
Bogomolov–Miyaoka–Yau inequality sets boundaries in the search for complex surfaces. Mapping out the topological types that are realized as complex surfaces is called
444:
1058:
1465:
1391:
1539:
450:
on the second cohomology, the
Bogomolov–Miyaoka–Yau inequality can also be written as a restriction on the topological type of the surface of general type:
275:
is a quotient of a ball. The latter statement is a consequence of Yau's differential geometric approach which is based on his resolution of the
100:. Its major interest is the way it restricts the possible topological types of the underlying real 4-manifold. It was proved independently by
945:
915:
1233:
1594:
447:
1537:
Yau, Shing Tung (1978), "On the Ricci curvature of a compact Kähler manifold and the complex Monge-Ampère equation. I",
456:
338:
1584:
155:
343:
1589:
1335:
1279:
1168:
97:
586:
220:
516:
37:
1599:
733:
673:
582:
617:
1474:
1400:
1389:
Van de Ven, Antonius (1966), "On the Chern numbers of certain complex and almost complex manifolds",
1354:
1298:
1177:
933:
910:, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge., vol. 4, Springer-Verlag, Berlin,
334:
139:
285:
1500:
1426:
1378:
1344:
1322:
1288:
1258:
1209:
1101:
1067:
1056:
Easton, Robert W. (2008), "Surfaces violating
Bogomolov-Miyaoka-Yau in positive characteristic",
998:
578:
701:
by an infinite discrete group. Examples of surfaces satisfying this equality are hard to find.
420:
1556:
1526:
1492:
1452:
1418:
1250:
1193:
1131:
1085:
1017:
974:
941:
911:
276:
93:
1548:
1516:
1482:
1442:
1408:
1362:
1306:
1242:
1185:
1121:
1112:
Ishida, Masa-Nori (1988), "An elliptic surface covered by
Mumford's fake projective plane",
1077:
1045:
1007:
1568:
1512:
1438:
1374:
1318:
1270:
1205:
1159:
1143:
1097:
1036:
Cartwright, Donald I.; Steger, Tim (2010), "Enumeration of the 50 fake projective planes",
1029:
986:
955:
925:
1564:
1508:
1463:
Yau, Shing Tung (1977), "Calabi's conjecture and some new results in algebraic geometry",
1434:
1370:
1314:
1266:
1201:
1155:
1139:
1093:
1025:
982:
962:
951:
921:
125:
1478:
1404:
1358:
1302:
1181:
166:
The conventional formulation of the
Bogomolov–Miyaoka–Yau inequality is as follows. Let
113:
101:
1521:
1447:
17:
1578:
1224:
1220:
1213:
1012:
725:
90:
1382:
1150:
Lang, William E. (1983), "Examples of surfaces of general type with vector fields",
1105:
1326:
993:
171:
135:
86:
1081:
211:
1466:
Proceedings of the
National Academy of Sciences of the United States of America
1392:
Proceedings of the
National Academy of Sciences of the United States of America
1154:, Progr. Math., vol. 36, Boston, MA: Birkhäuser Boston, pp. 167–173,
1049:
906:
Barth, Wolf P.; Hulek, Klaus; Peters, Chris A.M.; Van de Ven, Antonius (2004),
1366:
1333:
Prasad, Gopal; Yeung, Sai-Kee (2010), "Addendum to "Fake projective planes"",
1310:
1560:
1496:
1422:
1254:
1197:
1135:
1126:
1089:
1021:
978:
1166:
Miyaoka, Yoichi (1977), "On the Chern numbers of surfaces of general type",
1552:
1530:
1487:
1456:
1413:
838:= 45, and taking unbranched coverings of this quotient gives examples with
792:
gave a method for finding examples, which in particular produced a surface
965:(1978), "Holomorphic tensors and vector bundles on projective manifolds",
662:, so that equality holds in the Bogomolov–Miyaoka–Yau inequality, then
1262:
1189:
940:, Aspects of Mathematics, D4, Braunschweig: Friedr. Vieweg & Sohn,
1504:
1430:
1293:
1072:
1246:
1349:
132:) proved weaker versions with the constant 3 replaced by 8 and 4.
1277:
Prasad, Gopal; Yeung, Sai-Kee (2007), "Fake projective planes",
996:(1963), "Compact Clifford-Klein forms of symmetric spaces",
786:) showed that there are exactly 50 fake projective planes.
865:. Donald I. Cartwright and Tim Steger (
782:, Donald I. Cartwright and Tim Steger (
789:
967:
Izvestiya
Akademii Nauk SSSR. Seriya Matematicheskaya
676:
620:
519:
459:
423:
346:
288:
223:
40:
214:
of the complex tangent bundle of the surface. Then
1241:(1), The Johns Hopkins University Press: 233–244,
693:
654:
566:
503:{\displaystyle \sigma (X)\leq {\frac {1}{3}}e(X),}
502:
438:
409:
325:
260:
74:
755:= 9, which is the minimum possible value because
1059:Proceedings of the American Mathematical Society
866:
783:
705:showed that there are infinitely many values of
670:is isomorphic to a quotient of the unit ball in
938:Geradenkonfigurationen und Algebraische Flächen
1540:Communications on Pure and Applied Mathematics
1473:(5), National Academy of Sciences: 1798–1799,
1399:(6), National Academy of Sciences: 1624–1627,
410:{\displaystyle c_{1}^{2}(X)=2e(X)+3\sigma (X)}
150:) gave examples of surfaces in characteristic
8:
1225:"An algebraic surface with K ample, (K)=9, p
779:
775:
121:
1520:
1486:
1446:
1412:
1348:
1292:
1125:
1071:
1011:
685:
680:
679:
678:
675:
646:
630:
625:
619:
541:
518:
475:
458:
422:
356:
351:
345:
293:
287:
249:
233:
228:
222:
129:
66:
50:
45:
39:
729:
574:then the universal covering is a ball.
120:), after Antonius Van de Ven (
117:
819:found a quotient of this surface with
816:
790:Barthel, Hirzebruch & Höfer (1987)
147:
702:
261:{\displaystyle c_{1}^{2}\leq 3c_{2}.}
7:
567:{\displaystyle \sigma (X)=(1/3)e(X)}
143:
75:{\displaystyle c_{1}^{2}\leq 3c_{2}}
663:
109:
105:
694:{\displaystyle {\mathbb {C} }^{2}}
614:is a surface of general type with
25:
1044:(1), Elsevier Masson SAS: 11–13,
339:Thom–Hirzebruch signature theorem
146:) and Robert W. Easton (
1152:Arithmetic and geometry, Vol. II
655:{\displaystyle c_{1}^{2}=3c_{2}}
271:Moreover if equality holds then
170:be a compact complex surface of
29:Bogomolov–Miyaoka–Yau inequality
1234:American Journal of Mathematics
1114:The Tohoku Mathematical Journal
774:is always divisible by 12, and
561:
555:
549:
535:
529:
523:
494:
488:
469:
463:
433:
427:
404:
398:
386:
380:
368:
362:
320:
314:
305:
299:
1:
1082:10.1090/S0002-9939-08-09466-5
326:{\displaystyle c_{2}(X)=e(X)}
162:Formulation of the inequality
1013:10.1016/0040-9383(63)90026-0
724:for which a surface exists.
156:generalized Raynaud surfaces
1038:Comptes Rendus Mathématique
892:for every positive integer
1616:
1050:10.1016/j.crma.2009.11.016
439:{\displaystyle \sigma (X)}
210:) be the first and second
1367:10.1007/s00222-010-0259-6
1311:10.1007/s00222-007-0034-5
861:for any positive integer
780:Prasad & Yeung (2010)
776:Prasad & Yeung (2007)
1336:Inventiones Mathematicae
1280:Inventiones Mathematicae
1169:Inventiones Mathematicae
936:; Höfer, Thomas (1987),
908:Compact Complex Surfaces
587:surfaces of general type
446:is the signature of the
1553:10.1002/cpa.3160310304
1488:10.1073/pnas.74.5.1798
1414:10.1073/pnas.55.6.1624
1127:10.2748/tmj/1178227980
869:) found examples with
695:
656:
568:
504:
440:
411:
327:
262:
158:, for which it fails.
76:
18:Miyaoka–Yau inequality
1595:Differential geometry
934:Hirzebruch, Friedrich
734:fake projective plane
696:
657:
583:geography of surfaces
569:
505:
441:
412:
328:
263:
77:
932:Barthel, Gottfried;
674:
618:
517:
457:
421:
344:
335:Euler characteristic
286:
221:
140:Friedrich Hirzebruch
38:
27:In mathematics, the
1479:1977PNAS...74.1798Y
1405:1966PNAS...55.1624V
1359:2010InMat.182..213P
1303:2007InMat.168..321P
1182:1977InMat..42..225M
963:Bogomolov, Fedor A.
635:
361:
333:is the topological
238:
126:Fedor Bogomolov
55:
1585:Algebraic surfaces
1190:10.1007/BF01389789
691:
652:
621:
579:Noether inequality
577:Together with the
564:
500:
436:
407:
347:
323:
258:
224:
114:Yoichi Miyaoka
102:Shing-Tung Yau
72:
41:
31:is the inequality
1116:, Second Series,
947:978-3-528-08907-8
917:978-3-540-00832-3
726:David Mumford
483:
448:intersection form
277:Calabi conjecture
16:(Redirected from
1607:
1590:Complex surfaces
1571:
1533:
1524:
1490:
1459:
1450:
1416:
1385:
1352:
1329:
1296:
1273:
1216:
1162:
1146:
1129:
1108:
1075:
1066:(7): 2271–2278,
1052:
1032:
1015:
1006:(1–2): 111–122,
989:
973:(6): 1227–1287,
958:
928:
880:
879:
849:
848:
830:
829:
807:
806:
766:
765:
747:
746:
716:
715:
700:
698:
697:
692:
690:
689:
684:
683:
661:
659:
658:
653:
651:
650:
634:
629:
573:
571:
570:
565:
545:
509:
507:
506:
501:
484:
476:
445:
443:
442:
437:
416:
414:
413:
408:
360:
355:
332:
330:
329:
324:
298:
297:
267:
265:
264:
259:
254:
253:
237:
232:
94:complex surfaces
81:
79:
78:
73:
71:
70:
54:
49:
21:
1615:
1614:
1610:
1609:
1608:
1606:
1605:
1604:
1575:
1574:
1536:
1462:
1388:
1332:
1276:
1247:10.2307/2373947
1228:
1219:
1165:
1149:
1111:
1055:
1035:
992:
961:
948:
931:
918:
905:
902:
887:
878:
875:
874:
873:
856:
847:
844:
843:
842:
837:
828:
825:
824:
823:
814:
805:
802:
801:
800:
773:
764:
761:
760:
759:
754:
745:
742:
741:
740:
723:
714:
711:
710:
709:
677:
672:
671:
642:
616:
615:
608:
606:
599:
515:
514:
455:
454:
419:
418:
342:
341:
289:
284:
283:
245:
219:
218:
205:
198:
187:
180:
164:
62:
36:
35:
23:
22:
15:
12:
11:
5:
1613:
1611:
1603:
1602:
1597:
1592:
1587:
1577:
1576:
1573:
1572:
1547:(3): 339–411,
1534:
1460:
1386:
1343:(1): 213–227,
1330:
1287:(2): 321–370,
1274:
1226:
1221:Mumford, David
1217:
1176:(1): 225–237,
1163:
1147:
1120:(3): 367–396,
1109:
1053:
1033:
990:
959:
946:
929:
916:
901:
898:
885:
876:
854:
845:
835:
826:
812:
803:
771:
762:
752:
743:
721:
712:
688:
682:
649:
645:
641:
638:
633:
628:
624:
607:
604:
597:
593:Surfaces with
591:
563:
560:
557:
554:
551:
548:
544:
540:
537:
534:
531:
528:
525:
522:
511:
510:
499:
496:
493:
490:
487:
482:
479:
474:
471:
468:
465:
462:
435:
432:
429:
426:
406:
403:
400:
397:
394:
391:
388:
385:
382:
379:
376:
373:
370:
367:
364:
359:
354:
350:
322:
319:
316:
313:
310:
307:
304:
301:
296:
292:
269:
268:
257:
252:
248:
244:
241:
236:
231:
227:
203:
196:
185:
178:
163:
160:
83:
82:
69:
65:
61:
58:
53:
48:
44:
24:
14:
13:
10:
9:
6:
4:
3:
2:
1612:
1601:
1598:
1596:
1593:
1591:
1588:
1586:
1583:
1582:
1580:
1570:
1566:
1562:
1558:
1554:
1550:
1546:
1542:
1541:
1535:
1532:
1528:
1523:
1518:
1514:
1510:
1506:
1502:
1498:
1494:
1489:
1484:
1480:
1476:
1472:
1468:
1467:
1461:
1458:
1454:
1449:
1444:
1440:
1436:
1432:
1428:
1424:
1420:
1415:
1410:
1406:
1402:
1398:
1394:
1393:
1387:
1384:
1380:
1376:
1372:
1368:
1364:
1360:
1356:
1351:
1346:
1342:
1338:
1337:
1331:
1328:
1324:
1320:
1316:
1312:
1308:
1304:
1300:
1295:
1290:
1286:
1282:
1281:
1275:
1272:
1268:
1264:
1260:
1256:
1252:
1248:
1244:
1240:
1236:
1235:
1230:
1222:
1218:
1215:
1211:
1207:
1203:
1199:
1195:
1191:
1187:
1183:
1179:
1175:
1171:
1170:
1164:
1161:
1157:
1153:
1148:
1145:
1141:
1137:
1133:
1128:
1123:
1119:
1115:
1110:
1107:
1103:
1099:
1095:
1091:
1087:
1083:
1079:
1074:
1069:
1065:
1061:
1060:
1054:
1051:
1047:
1043:
1039:
1034:
1031:
1027:
1023:
1019:
1014:
1009:
1005:
1001:
1000:
995:
994:Borel, Armand
991:
988:
984:
980:
976:
972:
968:
964:
960:
957:
953:
949:
943:
939:
935:
930:
927:
923:
919:
913:
909:
904:
903:
899:
897:
895:
891:
884:
872:
868:
864:
860:
853:
841:
834:
822:
818:
817:Ishida (1988)
811:
799:
795:
791:
787:
785:
781:
777:
770:
758:
751:
739:
735:
731:
727:
720:
708:
704:
686:
669:
665:
647:
643:
639:
636:
631:
626:
622:
613:
603:
596:
592:
590:
588:
584:
580:
575:
558:
552:
546:
542:
538:
532:
526:
520:
497:
491:
485:
480:
477:
472:
466:
460:
453:
452:
451:
449:
430:
424:
401:
395:
392:
389:
383:
377:
374:
371:
365:
357:
352:
348:
340:
336:
317:
311:
308:
302:
294:
290:
280:
278:
274:
255:
250:
246:
242:
239:
234:
229:
225:
217:
216:
215:
213:
209:
202:
195:
191:
184:
177:
173:
169:
161:
159:
157:
153:
149:
145:
141:
137:
133:
131:
127:
123:
119:
115:
111:
107:
103:
99:
95:
92:
88:
87:Chern numbers
67:
63:
59:
56:
51:
46:
42:
34:
33:
32:
30:
19:
1600:Inequalities
1544:
1538:
1470:
1464:
1396:
1390:
1340:
1334:
1294:math/0512115
1284:
1278:
1238:
1232:
1173:
1167:
1151:
1117:
1113:
1073:math/0511455
1063:
1057:
1041:
1037:
1003:
997:
970:
966:
937:
907:
893:
889:
882:
870:
862:
858:
851:
839:
832:
820:
809:
797:
793:
788:
768:
756:
749:
737:
718:
706:
703:Borel (1963)
667:
666:proved that
611:
609:
601:
594:
576:
513:moreover if
512:
281:
272:
270:
207:
200:
193:
189:
182:
175:
172:general type
167:
165:
151:
136:Armand Borel
134:
98:general type
84:
28:
26:
337:and by the
212:Chern class
1579:Categories
900:References
732:) found a
664:Yau (1977)
174:, and let
154:, such as
1561:0010-3640
1497:0027-8424
1423:0027-8424
1350:0906.4932
1255:0002-9327
1214:120699065
1198:0020-9910
1136:0040-8735
1090:0002-9939
1022:0040-9383
979:0373-2436
521:σ
473:≤
461:σ
425:σ
396:σ
240:≤
57:≤
1531:16592394
1457:16578639
1383:17216453
1223:(1979),
1106:35276117
999:Topology
85:between
1569:0480350
1513:0451180
1475:Bibcode
1439:0198496
1401:Bibcode
1375:2672284
1355:Bibcode
1327:1990160
1319:2289867
1299:Bibcode
1271:0527834
1263:2373947
1206:0460343
1178:Bibcode
1160:0717611
1144:0957050
1098:2390492
1030:0146301
987:0522939
956:0912097
926:2030225
815:= 35.
728: (
199:=
181:=
128: (
116: (
104: (
91:compact
1567:
1559:
1529:
1522:431004
1519:
1511:
1503:
1495:
1455:
1448:224368
1445:
1437:
1429:
1421:
1381:
1373:
1325:
1317:
1269:
1261:
1253:
1212:
1204:
1196:
1158:
1142:
1134:
1104:
1096:
1088:
1028:
1020:
985:
977:
954:
944:
924:
914:
585:. see
417:where
282:Since
192:) and
124:) and
112:) and
1505:67110
1501:JSTOR
1431:57245
1427:JSTOR
1379:S2CID
1345:arXiv
1323:S2CID
1289:arXiv
1259:JSTOR
1229:=q=0"
1210:S2CID
1102:S2CID
1068:arXiv
796:with
736:with
1557:ISSN
1527:PMID
1493:ISSN
1453:PMID
1419:ISSN
1251:ISSN
1194:ISSN
1132:ISSN
1086:ISSN
1018:ISSN
975:ISSN
942:ISBN
912:ISBN
867:2010
857:= 45
784:2010
730:1979
148:2008
144:1983
138:and
130:1978
122:1966
118:1977
110:1978
106:1977
1549:doi
1517:PMC
1483:doi
1443:PMC
1409:doi
1363:doi
1341:182
1307:doi
1285:168
1243:doi
1239:101
1186:doi
1122:doi
1078:doi
1064:136
1046:doi
1042:348
1008:doi
888:= 9
881:= 3
850:= 3
831:= 3
808:= 3
748:= 3
717:= 3
610:If
600:= 3
96:of
89:of
1581::
1565:MR
1563:,
1555:,
1545:31
1543:,
1525:,
1515:,
1509:MR
1507:,
1499:,
1491:,
1481:,
1471:74
1469:,
1451:,
1441:,
1435:MR
1433:,
1425:,
1417:,
1407:,
1397:55
1395:,
1377:,
1371:MR
1369:,
1361:,
1353:,
1339:,
1321:,
1315:MR
1313:,
1305:,
1297:,
1283:,
1267:MR
1265:,
1257:,
1249:,
1237:,
1231:,
1208:,
1202:MR
1200:,
1192:,
1184:,
1174:42
1172:,
1156:MR
1140:MR
1138:,
1130:,
1118:40
1100:,
1094:MR
1092:,
1084:,
1076:,
1062:,
1040:,
1026:MR
1024:,
1016:,
1002:,
983:MR
981:,
971:42
969:,
952:MR
950:,
922:MR
920:,
896:.
778:,
767:+
589:.
279:.
108:,
1551::
1485::
1477::
1411::
1403::
1365::
1357::
1347::
1309::
1301::
1291::
1245::
1227:g
1188::
1180::
1124::
1080::
1070::
1048::
1010::
1004:2
894:n
890:n
886:2
883:c
877:1
871:c
863:k
859:k
855:2
852:c
846:1
840:c
836:2
833:c
827:1
821:c
813:2
810:c
804:1
798:c
794:X
772:2
769:c
763:1
757:c
753:2
750:c
744:1
738:c
722:2
719:c
713:1
707:c
687:2
681:C
668:X
648:2
644:c
640:3
637:=
632:2
627:1
623:c
612:X
605:2
602:c
598:1
595:c
562:)
559:X
556:(
553:e
550:)
547:3
543:/
539:1
536:(
533:=
530:)
527:X
524:(
498:,
495:)
492:X
489:(
486:e
481:3
478:1
470:)
467:X
464:(
434:)
431:X
428:(
405:)
402:X
399:(
393:3
390:+
387:)
384:X
381:(
378:e
375:2
372:=
369:)
366:X
363:(
358:2
353:1
349:c
321:)
318:X
315:(
312:e
309:=
306:)
303:X
300:(
295:2
291:c
273:X
256:.
251:2
247:c
243:3
235:2
230:1
226:c
208:X
206:(
204:2
201:c
197:2
194:c
190:X
188:(
186:1
183:c
179:1
176:c
168:X
152:p
68:2
64:c
60:3
52:2
47:1
43:c
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.