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