79:
The essential definition of stability for principal bundles was made by
Ramanathan, but applies only to the case of Riemann surfaces. In this section we state the definition as appearing in the work of Anchouche and Biswas which is valid over any Kähler manifold, and indeed makes sense more generally
2258:-Higgs bundle. The above definition of stability for principal bundles generalises to these objects by requiring the reductions of structure group are compatible with the Higgs field of the principal Higgs bundle. It was shown by Anchouche and Biswas that the analogue of the
534:
1362:
813:
1840:
2355:
Ramanathan, A., 1996, November. Moduli for principal bundles over algebraic curves: II. In
Proceedings of the Indian Academy of Sciences-Mathematical Sciences (Vol. 106, No. 4, pp. 421-449). Springer India.
2346:
Ramanathan, A., 1996, August. Moduli for principal bundles over algebraic curves: I. In
Proceedings of the Indian Academy of Sciences-Mathematical Sciences (Vol. 106, No. 3, pp. 301-328). Springer India.
1452:
450:
2164:
1916:
1218:
1262:
1493:
575:
1628:
629:
2380:
Anchouche, B. and Biswas, I., 2001. Einstein-Hermitian connections on polystable principal bundles over a compact Kähler manifold. American
Journal of Mathematics, 123(2), pp.207-228.
686:
67:
if and only if it is polystable, was shown to be true in the case of projective manifolds by
Subramanian and Ramanathan, and for arbitrary compact Kähler manifolds by Anchouche and
2104:
2058:
1965:
1291:
1121:
350:
2212:
1992:
1667:
137:
2312:
169:
2130:
1713:
406:
376:
991:
852:
724:
215:
458:
1863:
1061:
2280:
2256:
2188:
2078:
2032:
2012:
1936:
1753:
1733:
1687:
1593:
1573:
1553:
1533:
1513:
1406:
1386:
1169:
1149:
1081:
1035:
1011:
956:
936:
916:
896:
876:
299:
275:
255:
235:
189:
105:
55:
Many statements about the stability of vector bundles can be translated into the language of stable principal bundles. For example, the analogue of the
1299:
732:
1761:
2226:
of the adjoint bundle (because a representation that splits as a direct sum would lead to the associated bundle splitting as a direct sum).
56:
1411:
411:
64:
2259:
632:
2406:
2135:
1875:
1177:
1227:
1457:
2334:
Ramanathan, A., 1975. Stable principal bundles on a compact
Riemann surface. Mathematische Annalen, 213(2), pp.129-152.
2401:
2219:
1365:
542:
310:
1598:
584:
638:
2364:
Subramanian, S. and
Ramanathan, A., 1988. Einstein-Hermitian connections on principal bundles and stability.
2215:
2396:
2167:
2083:
2037:
1944:
21:
1267:
1086:
315:
2193:
1973:
1866:
1633:
1221:
41:
33:
1388:, and the above definition of stability of the principal bundle is equivalent to slope stability of
81:
120:
2315:
2285:
142:
25:
2109:
1692:
385:
355:
60:
1294:
964:
821:
529:{\displaystyle \deg \sigma ^{*}T_{\operatorname {rel} }P/Q>0\quad ({\text{resp. }}\geq 0).}
278:
703:
194:
37:
1848:
1014:
114:
111:
108:
68:
45:
1040:
84:. This reduces to Ramanathan's definition in the case the manifold is a Riemann surface.
2166:
and can take the associated subbundle. In this case more care must be taken because the
2265:
2241:
2173:
2063:
2017:
1997:
1939:
1921:
1738:
1718:
1672:
1578:
1558:
1538:
1518:
1498:
1391:
1371:
1154:
1134:
1066:
1020:
996:
941:
921:
901:
881:
861:
697:
578:
379:
284:
260:
240:
220:
174:
90:
2222:, the latter condition hinting at why stability of the principal bundle only leads to
2390:
693:
49:
2235:
1715:
on which the reduction of structure group is defined, and therefore a subsheaf of
1063:, so it is enough to consider reductions of structure group over the entirety of
1968:
1357:{\displaystyle E=P\times _{\operatorname {GL} (n,\mathbb {C} )}\mathbb {C} ^{n}}
855:
17:
2106:
is slope polystable. Again the key point here is that for a parabolic subgroup
938:
is bigger than two, the value of the integral will agree with that over all of
808:{\displaystyle \operatorname {deg} (F):=\int _{X}c_{1}(F)\wedge \omega ^{n-1},}
1835:{\displaystyle \deg \sigma ^{*}T_{\operatorname {rel} }P/Q=\mu (E)-\mu (F)}
44:
for the purpose of defining the moduli space of G-principal bundles over a
59:
for principal bundles, that a holomorphic principal bundle over a compact
878:. In the above setting the degree is computed for a bundle defined over
277:
vary holomorphically, which makes sense as the structure group is a
1171:
there are several natural vector bundles one may associate to it.
40:. The concept of stability for principal bundles was introduced by
2234:
Just as one can generalise a vector bundle to the notion of a
1447:{\displaystyle Q\subset \operatorname {GL} (n,\mathbb {C} )}
427:
1408:. The essential point is that a maximal parabolic subgroup
650:
590:
445:{\displaystyle \operatorname {codim} (X\backslash U)\geq 2}
257:. Holomorphic here means that the transition functions for
2238:, it is possible to formulate a definition of a principal
2159:{\displaystyle {\mathfrak {q}}\subset {\mathfrak {g}}}
2288:
2268:
2244:
2196:
2176:
2138:
2112:
2086:
2066:
2040:
2020:
2000:
1976:
1947:
1924:
1918:
there is still a natural associated vector bundle to
1911:{\displaystyle G=\operatorname {GL} (n,\mathbb {C} )}
1878:
1851:
1764:
1741:
1721:
1695:
1675:
1636:
1601:
1581:
1561:
1541:
1521:
1501:
1460:
1414:
1394:
1374:
1302:
1270:
1230:
1213:{\displaystyle G=\operatorname {GL} (n,\mathbb {C} )}
1180:
1157:
1137:
1089:
1069:
1043:
1023:
999:
967:
944:
924:
904:
884:
864:
824:
735:
706:
641:
587:
545:
461:
414:
388:
358:
318:
287:
263:
243:
223:
197:
177:
145:
123:
93:
2282:-Higgs bundles in the case where the base manifold
1257:{\displaystyle \operatorname {GL} (n,\mathbb {C} )}
52:and others on the moduli spaces of vector bundles.
2306:
2274:
2250:
2206:
2182:
2158:
2124:
2098:
2072:
2052:
2026:
2006:
1986:
1959:
1930:
1910:
1857:
1834:
1747:
1727:
1707:
1681:
1661:
1622:
1587:
1567:
1547:
1527:
1507:
1488:{\displaystyle 0\subset W\subset \mathbb {C} ^{n}}
1487:
1446:
1400:
1380:
1356:
1285:
1256:
1212:
1163:
1143:
1115:
1075:
1055:
1029:
1005:
985:
950:
930:
910:
890:
870:
846:
807:
718:
680:
623:
569:
528:
444:
400:
370:
344:
293:
269:
249:
229:
209:
183:
171:be a compact Kähler manifold of complex dimension
163:
131:
99:
918:, but since the codimension of the complement of
2034:is semistable if and only if the adjoint bundle
2262:for Higgs vector bundles is true for principal
8:
570:{\displaystyle T_{\operatorname {rel} }P/Q}
2287:
2267:
2243:
2198:
2197:
2195:
2175:
2150:
2149:
2140:
2139:
2137:
2111:
2085:
2065:
2039:
2019:
1999:
1978:
1977:
1975:
1946:
1923:
1901:
1900:
1877:
1850:
1794:
1785:
1775:
1763:
1740:
1720:
1694:
1674:
1650:
1635:
1623:{\displaystyle W\subset \mathbb {C} ^{n}}
1614:
1610:
1609:
1600:
1580:
1560:
1540:
1520:
1500:
1479:
1475:
1474:
1459:
1437:
1436:
1413:
1393:
1373:
1348:
1344:
1343:
1333:
1332:
1316:
1301:
1277:
1273:
1272:
1269:
1247:
1246:
1229:
1203:
1202:
1179:
1156:
1136:
1105:
1088:
1068:
1042:
1022:
998:
966:
943:
923:
903:
883:
863:
829:
823:
790:
768:
758:
734:
705:
669:
656:
640:
624:{\displaystyle \left.P/Q\right|_{U}\to U}
609:
596:
586:
559:
550:
544:
509:
491:
482:
472:
460:
413:
408:is some open subset with the codimension
387:
357:
334:
317:
286:
262:
242:
222:
196:
176:
144:
125:
124:
122:
92:
2376:
2374:
2372:
2370:
2060:is slope semistable, and furthermore if
2327:
1127:Relation to stability of vector bundles
681:{\displaystyle T(\left.P/Q\right|_{U})}
32:is a generalisation of the notion of a
2342:
2340:
1224:, then the standard representation of
1017:, by assumption on the codimension of
48:, a generalisation of earlier work by
2132:, one obtains a parabolic subalgebra
1630:, one may take the associated bundle
7:
2099:{\displaystyle \operatorname {ad} P}
2053:{\displaystyle \operatorname {ad} P}
1960:{\displaystyle \operatorname {ad} P}
2199:
2151:
2141:
1979:
14:
1872:When the structure group is not
1515:is invariant under the subgroup
1454:corresponds to a choice of flag
1286:{\displaystyle \mathbb {C} ^{n}}
1151:-bundle for a complex Lie group
1116:{\displaystyle \sigma :X\to P/Q}
345:{\displaystyle \sigma :U\to P/Q}
57:Kobayashi–Hitchin correspondence
2260:nonabelian Hodge correspondence
2207:{\displaystyle {\mathfrak {g}}}
1987:{\displaystyle {\mathfrak {g}}}
1755:. It can then be computed that
1662:{\displaystyle F=P\times _{Q}W}
1535:. Since the structure group of
505:
2301:
2289:
1905:
1891:
1829:
1823:
1814:
1808:
1595:preserves the vector subspace
1441:
1427:
1337:
1323:
1251:
1237:
1207:
1193:
1099:
961:Notice that in the case where
841:
835:
780:
774:
748:
742:
710:
675:
645:
615:
520:
506:
433:
421:
328:
201:
158:
146:
1:
1293:allows one to construct the
311:reduction of structure group
132:{\displaystyle \mathbb {C} }
2307:{\displaystyle (X,\omega )}
1669:, which is a sub-bundle of
217:is a holomorphic principal
164:{\displaystyle (X,\omega )}
65:Hermite–Einstein connection
2423:
2316:complex projective variety
2125:{\displaystyle Q\subset G}
1967:, with fibre given by the
1708:{\displaystyle U\subset X}
401:{\displaystyle U\subset X}
371:{\displaystyle Q\subset G}
1366:holomorphic vector bundle
117:over the complex numbers
986:{\displaystyle \dim X=1}
847:{\displaystyle c_{1}(F)}
2014:. The principal bundle
1869:of the vector bundles.
631:otherwise known as the
281:. The principal bundle
30:stable principal bundle
2308:
2276:
2252:
2208:
2184:
2168:adjoint representation
2160:
2126:
2100:
2074:
2054:
2028:
2008:
1988:
1961:
1932:
1912:
1859:
1836:
1749:
1729:
1709:
1683:
1663:
1624:
1589:
1569:
1549:
1529:
1509:
1489:
1448:
1402:
1382:
1358:
1287:
1258:
1214:
1165:
1145:
1117:
1077:
1057:
1031:
1007:
987:
952:
932:
912:
892:
872:
848:
809:
720:
719:{\displaystyle F\to X}
682:
625:
571:
530:
446:
402:
372:
346:
295:
271:
251:
231:
211:
210:{\displaystyle P\to X}
185:
165:
133:
101:
2407:Differential geometry
2309:
2277:
2253:
2209:
2185:
2161:
2127:
2101:
2075:
2055:
2029:
2009:
1989:
1962:
1933:
1913:
1860:
1837:
1750:
1730:
1710:
1684:
1664:
1625:
1590:
1570:
1550:
1530:
1510:
1490:
1449:
1403:
1383:
1359:
1288:
1259:
1215:
1166:
1146:
1118:
1078:
1058:
1032:
1008:
988:
953:
933:
913:
893:
873:
849:
810:
721:
683:
626:
572:
531:
447:
403:
373:
347:
296:
272:
252:
232:
212:
186:
166:
134:
102:
22:differential geometry
2286:
2266:
2242:
2194:
2174:
2136:
2110:
2084:
2064:
2038:
2018:
1998:
1974:
1945:
1922:
1876:
1858:{\displaystyle \mu }
1849:
1762:
1739:
1719:
1693:
1673:
1634:
1599:
1579:
1559:
1555:has been reduced to
1539:
1519:
1499:
1458:
1412:
1392:
1372:
1300:
1268:
1228:
1222:general linear group
1178:
1155:
1135:
1087:
1067:
1041:
1021:
997:
965:
942:
922:
902:
882:
862:
822:
733:
704:
639:
585:
581:of the fibre bundle
543:
459:
412:
386:
356:
316:
285:
261:
241:
221:
195:
175:
143:
121:
91:
42:Annamalai Ramanathan
34:stable vector bundle
1056:{\displaystyle U=X}
82:algebraic varieties
2402:Algebraic geometry
2304:
2272:
2248:
2204:
2180:
2156:
2122:
2096:
2070:
2050:
2024:
2004:
1984:
1957:
1928:
1908:
1855:
1832:
1745:
1725:
1705:
1679:
1659:
1620:
1585:
1565:
1545:
1525:
1505:
1485:
1444:
1398:
1378:
1354:
1283:
1254:
1210:
1161:
1141:
1131:Given a principal
1113:
1073:
1053:
1037:we must have that
1027:
1003:
983:
948:
928:
908:
888:
868:
844:
805:
726:is defined to be
716:
688:. Recall that the
678:
621:
567:
526:
442:
398:
380:parabolic subgroup
368:
342:
291:
267:
247:
227:
207:
181:
161:
129:
97:
36:to the setting of
26:algebraic geometry
2275:{\displaystyle G}
2251:{\displaystyle G}
2183:{\displaystyle G}
2073:{\displaystyle P}
2027:{\displaystyle P}
2007:{\displaystyle G}
1931:{\displaystyle P}
1748:{\displaystyle X}
1728:{\displaystyle E}
1682:{\displaystyle E}
1588:{\displaystyle Q}
1568:{\displaystyle Q}
1548:{\displaystyle P}
1528:{\displaystyle Q}
1508:{\displaystyle W}
1401:{\displaystyle E}
1381:{\displaystyle X}
1295:associated bundle
1164:{\displaystyle G}
1144:{\displaystyle G}
1076:{\displaystyle X}
1030:{\displaystyle U}
1006:{\displaystyle X}
951:{\displaystyle X}
931:{\displaystyle U}
911:{\displaystyle X}
891:{\displaystyle U}
871:{\displaystyle F}
512:
294:{\displaystyle P}
279:complex Lie group
270:{\displaystyle P}
250:{\displaystyle X}
230:{\displaystyle G}
184:{\displaystyle n}
100:{\displaystyle G}
38:principal bundles
20:, and especially
2414:
2381:
2378:
2365:
2362:
2356:
2353:
2347:
2344:
2335:
2332:
2313:
2311:
2310:
2305:
2281:
2279:
2278:
2273:
2257:
2255:
2254:
2249:
2213:
2211:
2210:
2205:
2203:
2202:
2189:
2187:
2186:
2181:
2165:
2163:
2162:
2157:
2155:
2154:
2145:
2144:
2131:
2129:
2128:
2123:
2105:
2103:
2102:
2097:
2080:is stable, then
2079:
2077:
2076:
2071:
2059:
2057:
2056:
2051:
2033:
2031:
2030:
2025:
2013:
2011:
2010:
2005:
1993:
1991:
1990:
1985:
1983:
1982:
1966:
1964:
1963:
1958:
1937:
1935:
1934:
1929:
1917:
1915:
1914:
1909:
1904:
1864:
1862:
1861:
1856:
1841:
1839:
1838:
1833:
1798:
1790:
1789:
1780:
1779:
1754:
1752:
1751:
1746:
1734:
1732:
1731:
1726:
1714:
1712:
1711:
1706:
1689:over the subset
1688:
1686:
1685:
1680:
1668:
1666:
1665:
1660:
1655:
1654:
1629:
1627:
1626:
1621:
1619:
1618:
1613:
1594:
1592:
1591:
1586:
1574:
1572:
1571:
1566:
1554:
1552:
1551:
1546:
1534:
1532:
1531:
1526:
1514:
1512:
1511:
1506:
1494:
1492:
1491:
1486:
1484:
1483:
1478:
1453:
1451:
1450:
1445:
1440:
1407:
1405:
1404:
1399:
1387:
1385:
1384:
1379:
1363:
1361:
1360:
1355:
1353:
1352:
1347:
1341:
1340:
1336:
1292:
1290:
1289:
1284:
1282:
1281:
1276:
1263:
1261:
1260:
1255:
1250:
1219:
1217:
1216:
1211:
1206:
1170:
1168:
1167:
1162:
1150:
1148:
1147:
1142:
1122:
1120:
1119:
1114:
1109:
1082:
1080:
1079:
1074:
1062:
1060:
1059:
1054:
1036:
1034:
1033:
1028:
1012:
1010:
1009:
1004:
993:, that is where
992:
990:
989:
984:
957:
955:
954:
949:
937:
935:
934:
929:
917:
915:
914:
909:
897:
895:
894:
889:
877:
875:
874:
869:
853:
851:
850:
845:
834:
833:
814:
812:
811:
806:
801:
800:
773:
772:
763:
762:
725:
723:
722:
717:
687:
685:
684:
679:
674:
673:
668:
664:
660:
630:
628:
627:
622:
614:
613:
608:
604:
600:
577:is the relative
576:
574:
573:
568:
563:
555:
554:
535:
533:
532:
527:
513:
510:
495:
487:
486:
477:
476:
451:
449:
448:
443:
407:
405:
404:
399:
377:
375:
374:
369:
351:
349:
348:
343:
338:
300:
298:
297:
292:
276:
274:
273:
268:
256:
254:
253:
248:
236:
234:
233:
228:
216:
214:
213:
208:
190:
188:
187:
182:
170:
168:
167:
162:
138:
136:
135:
130:
128:
106:
104:
103:
98:
2422:
2421:
2417:
2416:
2415:
2413:
2412:
2411:
2387:
2386:
2385:
2384:
2379:
2368:
2363:
2359:
2354:
2350:
2345:
2338:
2333:
2329:
2324:
2284:
2283:
2264:
2263:
2240:
2239:
2232:
2230:Generalisations
2192:
2191:
2172:
2171:
2134:
2133:
2108:
2107:
2082:
2081:
2062:
2061:
2036:
2035:
2016:
2015:
1996:
1995:
1972:
1971:
1943:
1942:
1920:
1919:
1874:
1873:
1847:
1846:
1781:
1771:
1760:
1759:
1737:
1736:
1717:
1716:
1691:
1690:
1671:
1670:
1646:
1632:
1631:
1608:
1597:
1596:
1577:
1576:
1557:
1556:
1537:
1536:
1517:
1516:
1497:
1496:
1473:
1456:
1455:
1410:
1409:
1390:
1389:
1370:
1369:
1342:
1312:
1298:
1297:
1271:
1266:
1265:
1226:
1225:
1176:
1175:
1153:
1152:
1133:
1132:
1129:
1085:
1084:
1065:
1064:
1039:
1038:
1019:
1018:
1015:Riemann surface
995:
994:
963:
962:
940:
939:
920:
919:
900:
899:
880:
879:
860:
859:
825:
820:
819:
786:
764:
754:
731:
730:
702:
701:
652:
649:
648:
637:
636:
633:vertical bundle
592:
589:
588:
583:
582:
546:
541:
540:
478:
468:
457:
456:
410:
409:
384:
383:
354:
353:
314:
313:
309:) if for every
283:
282:
259:
258:
239:
238:
219:
218:
193:
192:
173:
172:
141:
140:
119:
118:
115:algebraic group
89:
88:
77:
61:Kähler manifold
46:Riemann surface
12:
11:
5:
2420:
2418:
2410:
2409:
2404:
2399:
2389:
2388:
2383:
2382:
2366:
2357:
2348:
2336:
2326:
2325:
2323:
2320:
2303:
2300:
2297:
2294:
2291:
2271:
2247:
2231:
2228:
2214:is not always
2201:
2179:
2153:
2148:
2143:
2121:
2118:
2115:
2095:
2092:
2089:
2069:
2049:
2046:
2043:
2023:
2003:
1981:
1956:
1953:
1950:
1940:adjoint bundle
1927:
1907:
1903:
1899:
1896:
1893:
1890:
1887:
1884:
1881:
1854:
1843:
1842:
1831:
1828:
1825:
1822:
1819:
1816:
1813:
1810:
1807:
1804:
1801:
1797:
1793:
1788:
1784:
1778:
1774:
1770:
1767:
1744:
1724:
1704:
1701:
1698:
1678:
1658:
1653:
1649:
1645:
1642:
1639:
1617:
1612:
1607:
1604:
1584:
1564:
1544:
1524:
1504:
1482:
1477:
1472:
1469:
1466:
1463:
1443:
1439:
1435:
1432:
1429:
1426:
1423:
1420:
1417:
1397:
1377:
1351:
1346:
1339:
1335:
1331:
1328:
1325:
1322:
1319:
1315:
1311:
1308:
1305:
1280:
1275:
1253:
1249:
1245:
1242:
1239:
1236:
1233:
1209:
1205:
1201:
1198:
1195:
1192:
1189:
1186:
1183:
1160:
1140:
1128:
1125:
1112:
1108:
1104:
1101:
1098:
1095:
1092:
1072:
1052:
1049:
1046:
1026:
1002:
982:
979:
976:
973:
970:
947:
927:
907:
887:
867:
843:
840:
837:
832:
828:
816:
815:
804:
799:
796:
793:
789:
785:
782:
779:
776:
771:
767:
761:
757:
753:
750:
747:
744:
741:
738:
715:
712:
709:
698:coherent sheaf
677:
672:
667:
663:
659:
655:
651:
647:
644:
620:
617:
612:
607:
603:
599:
595:
591:
579:tangent bundle
566:
562:
558:
553:
549:
537:
536:
525:
522:
519:
516:
508:
504:
501:
498:
494:
490:
485:
481:
475:
471:
467:
464:
441:
438:
435:
432:
429:
426:
423:
420:
417:
397:
394:
391:
367:
364:
361:
341:
337:
333:
330:
327:
324:
321:
290:
266:
246:
226:
206:
203:
200:
180:
160:
157:
154:
151:
148:
127:
96:
76:
73:
13:
10:
9:
6:
4:
3:
2:
2419:
2408:
2405:
2403:
2400:
2398:
2397:Fiber bundles
2395:
2394:
2392:
2377:
2375:
2373:
2371:
2367:
2361:
2358:
2352:
2349:
2343:
2341:
2337:
2331:
2328:
2321:
2319:
2317:
2298:
2295:
2292:
2269:
2261:
2245:
2237:
2229:
2227:
2225:
2224:polystability
2221:
2217:
2177:
2169:
2146:
2119:
2116:
2113:
2093:
2090:
2087:
2067:
2047:
2044:
2041:
2021:
2001:
1970:
1954:
1951:
1948:
1941:
1925:
1897:
1894:
1888:
1885:
1882:
1879:
1870:
1868:
1852:
1826:
1820:
1817:
1811:
1805:
1802:
1799:
1795:
1791:
1786:
1782:
1776:
1772:
1768:
1765:
1758:
1757:
1756:
1742:
1722:
1702:
1699:
1696:
1676:
1656:
1651:
1647:
1643:
1640:
1637:
1615:
1605:
1602:
1582:
1562:
1542:
1522:
1502:
1480:
1470:
1467:
1464:
1461:
1433:
1430:
1424:
1421:
1418:
1415:
1395:
1375:
1367:
1349:
1329:
1326:
1320:
1317:
1313:
1309:
1306:
1303:
1296:
1278:
1243:
1240:
1234:
1231:
1223:
1199:
1196:
1190:
1187:
1184:
1181:
1172:
1158:
1138:
1126:
1124:
1110:
1106:
1102:
1096:
1093:
1090:
1070:
1050:
1047:
1044:
1024:
1016:
1000:
980:
977:
974:
971:
968:
959:
945:
925:
905:
885:
865:
857:
854:is the first
838:
830:
826:
802:
797:
794:
791:
787:
783:
777:
769:
765:
759:
755:
751:
745:
739:
736:
729:
728:
727:
713:
707:
699:
695:
694:vector bundle
691:
670:
665:
661:
657:
653:
642:
634:
618:
610:
605:
601:
597:
593:
580:
564:
560:
556:
551:
547:
523:
517:
514:
502:
499:
496:
492:
488:
483:
479:
473:
469:
465:
462:
455:
454:
453:
439:
436:
430:
424:
418:
415:
395:
392:
389:
381:
365:
362:
359:
339:
335:
331:
325:
322:
319:
312:
308:
304:
288:
280:
264:
244:
237:-bundle over
224:
204:
198:
178:
155:
152:
149:
116:
113:
110:
94:
85:
83:
74:
72:
70:
66:
62:
58:
53:
51:
50:David Mumford
47:
43:
39:
35:
31:
27:
23:
19:
2360:
2351:
2330:
2236:Higgs bundle
2233:
2223:
1871:
1865:denotes the
1844:
1735:over all of
1364:. This is a
1173:
1130:
960:
817:
689:
538:
306:
302:
86:
78:
54:
29:
15:
2220:irreducible
1969:Lie algebra
1174:Firstly if
856:Chern class
511:resp.
452:, we have
307:semi-stable
18:mathematics
2391:Categories
2322:References
378:a maximal
301:is called
191:. Suppose
75:Definition
2299:ω
2147:⊂
2117:⊂
2091:
2045:
1952:
1889:
1853:μ
1821:μ
1818:−
1806:μ
1777:∗
1773:σ
1769:
1700:⊂
1648:×
1606:⊂
1471:⊂
1465:⊂
1425:
1419:⊂
1321:
1314:×
1235:
1191:
1100:→
1091:σ
972:
795:−
788:ω
784:∧
756:∫
740:
711:→
616:→
515:≥
474:∗
470:σ
466:
437:≥
428:∖
419:
393:⊂
363:⊂
329:→
320:σ
202:→
156:ω
112:reductive
109:connected
63:admits a
2216:faithful
1495:, where
898:inside
305:(resp.
1938:, the
1845:where
1575:, and
1220:, the
818:where
690:degree
382:where
303:stable
139:. Let
69:Biswas
2314:is a
1867:slope
1368:over
1013:is a
692:of a
539:Here
416:codim
107:be a
696:(or
500:>
352:for
87:Let
80:for
28:, a
24:and
2218:or
2190:on
2170:of
1994:of
1787:rel
1766:deg
1264:on
969:dim
858:of
737:deg
635:of
552:rel
484:rel
463:deg
16:In
2393::
2369:^
2339:^
2318:.
2088:ad
2042:ad
1949:ad
1886:GL
1422:GL
1318:GL
1232:GL
1188:GL
1123:.
1083:,
958:.
752::=
700:)
71:.
2302:)
2296:,
2293:X
2290:(
2270:G
2246:G
2200:g
2178:G
2152:g
2142:q
2120:G
2114:Q
2094:P
2068:P
2048:P
2022:P
2002:G
1980:g
1955:P
1926:P
1906:)
1902:C
1898:,
1895:n
1892:(
1883:=
1880:G
1830:)
1827:F
1824:(
1815:)
1812:E
1809:(
1803:=
1800:Q
1796:/
1792:P
1783:T
1743:X
1723:E
1703:X
1697:U
1677:E
1657:W
1652:Q
1644:P
1641:=
1638:F
1616:n
1611:C
1603:W
1583:Q
1563:Q
1543:P
1523:Q
1503:W
1481:n
1476:C
1468:W
1462:0
1442:)
1438:C
1434:,
1431:n
1428:(
1416:Q
1396:E
1376:X
1350:n
1345:C
1338:)
1334:C
1330:,
1327:n
1324:(
1310:P
1307:=
1304:E
1279:n
1274:C
1252:)
1248:C
1244:,
1241:n
1238:(
1208:)
1204:C
1200:,
1197:n
1194:(
1185:=
1182:G
1159:G
1139:G
1111:Q
1107:/
1103:P
1097:X
1094::
1071:X
1051:X
1048:=
1045:U
1025:U
1001:X
981:1
978:=
975:X
946:X
926:U
906:X
886:U
866:F
842:)
839:F
836:(
831:1
827:c
803:,
798:1
792:n
781:)
778:F
775:(
770:1
766:c
760:X
749:)
746:F
743:(
714:X
708:F
676:)
671:U
666:|
662:Q
658:/
654:P
646:(
643:T
619:U
611:U
606:|
602:Q
598:/
594:P
565:Q
561:/
557:P
548:T
524:.
521:)
518:0
507:(
503:0
497:Q
493:/
489:P
480:T
440:2
434:)
431:U
425:X
422:(
396:X
390:U
366:G
360:Q
340:Q
336:/
332:P
326:U
323::
289:P
265:P
245:X
225:G
205:X
199:P
179:n
159:)
153:,
150:X
147:(
126:C
95:G
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