897:
429:
736:
1002:
278:
1479:
1664:
724:
1532:
1052:
566:
1799:
1223:
1847:
157:
1135:
1097:
1737:
613:
1902:
in complex analysis which admit "many" holomorphic functions from the complex numbers into themselves. It is known that a Stein manifold is elliptic if and only if it is
892:{\displaystyle H^{1}(X,{\mathcal {O}}_{X})\longrightarrow H^{1}(X,{\mathcal {O}}_{X}^{*})\longrightarrow H^{2}(X,\mathbb {Z} )\longrightarrow H^{2}(X,{\mathcal {O}}_{X})}
261:
507:
1318:
1600:
1411:
184:
1572:
1552:
1431:
1378:
1358:
1338:
1292:
1185:
1165:
527:
481:
457:
231:
204:
124:
100:
1671:
Related to the previous item, another equivalent and more topological definition in complex dimension 2 is the following: a Stein surface is a complex surface
1899:
2304:
Stein, Karl (1951), "Analytische
Funktionen mehrerer komplexer Veränderlichen zu vorgegebenen Periodizitätsmoduln und das zweite Cousinsche Problem",
909:
632:
2009:
1236:
These facts imply that a Stein manifold is a closed complex submanifold of complex space, whose complex structure is that of the
424:{\displaystyle {\bar {K}}=\left\{z\in X\,\left|\,|f(z)|\leq \sup _{w\in K}|f(w)|\ \forall f\in {\mathcal {O}}(X)\right.\right\},}
2137:
2115:
28:
1440:
1977:
1613:
2102:. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge / A Series of Modern Surveys in Mathematics. Vol. 56.
1864:
Numerous further characterizations of such manifolds exist, in particular capturing the property of their having "many"
665:
2229:
2088:
1484:
1951:
1930: = 2 the same holds provided the 2-handles are attached with certain framings (framing less than the
1876:. The initial impetus was to have a description of the properties of the domain of definition of the (maximal)
1961:
1931:
1007:
532:
1869:
903:
1956:
1934:). Every closed smooth 4-manifold is a union of two Stein 4-manifolds glued along their common boundary.
1753:
1190:
265:
17:
1804:
2388:
727:
129:
66:
is similar to a Stein manifold but is allowed to have singularities. Stein spaces are the analogues of
55:
1111:
1073:
1690:
1390:
655:
571:
2224:, Grundlehren der Mathematischen Wissenschaften, vol. 236, Berlin-New York: Springer-Verlag,
240:
2157:
1993:
1877:
486:
2246:
Ornea, Liviu; Verbitsky, Misha (2010). "Locally conformal Kähler manifolds with potential".
2132:, North-Holland Mathematical Library, vol. 7, Amsterdam: North-Holland Publishing Co.,
1297:
2371:
2325:
2239:
2204:
2147:
2016:
1865:
1585:
1396:
1105:
160:
8:
166:
2349:
2329:
2292:
2263:
2208:
2192:
2166:
2064:
2044:
1850:
1606:
to the idea of a corresponding class of compact complex manifolds with boundary called
1557:
1537:
1416:
1363:
1343:
1323:
1277:
1170:
1150:
512:
466:
442:
216:
189:
109:
85:
2333:
2225:
2184:
2133:
2111:
2084:
1881:
1579:
1147:
The embedding theorem for Stein manifolds states the following: Every Stein manifold
2267:
2212:
2125:
2068:
2059:
2032:
2359:
2313:
2284:
2255:
2176:
2103:
2054:
1973:
1873:
1740:
103:
32:
1360:
which form a local coordinate system when restricted to some open neighborhood of
2367:
2321:
2235:
2200:
2143:
2013:
1258:
1058:
636:
628:
2363:
651:
2005:
1903:
1892:
1262:
71:
67:
51:
2259:
2107:
1142:
Every closed complex submanifold of a Stein manifold is a Stein manifold, too.
2382:
2216:(definitions and constructions of Stein domains and manifolds in dimension 4)
2188:
1666:. Some authors call such manifolds therefore strictly pseudoconvex manifolds.
1266:
1237:
1226:
234:
2275:
Iss'Sa, Hej (1966). "On the
Meromorphic Function Field of a Stein Variety".
1976:, Topological characterization of Stein manifolds of dimension > 2,
1257:
In one complex dimension the Stein condition can be simplified: a connected
2076:
1989:
1434:
647:
44:
2083:, Graduate Text in Mathematics, vol. 81, New-York: Springer Verlag,
997:{\displaystyle H^{1}(X,{\mathcal {O}}_{X})=H^{2}(X,{\mathcal {O}}_{X})=0}
40:
2317:
2296:
1229:
2196:
2155:
Gompf, Robert E. (1998), "Handlebody construction of Stein surfaces",
2354:
2171:
2288:
2180:
2008:
and
Rostislav Matveyev, A convex decomposition for four-manifolds,
618:
2049:
2094:(including a proof of Behnke-Stein and Grauert–Röhrl theorems)
1265:
it is not compact. This can be proved using a version of the
1064:
662:
is trivial. In particular, every line bundle is trivial, so
1888:
1389:. The latter means that it has a strongly pseudoconvex (or
410:
2165:(2), The Annals of Mathematics, Vol. 148, No. 2: 619–693,
1385:
Being a Stein manifold is equivalent to being a (complex)
1906:
in the sense of so-called "holomorphic homotopy theory".
2130:
An introduction to complex analysis in several variables
1750:
agreeing with the usual orientation as the boundary of
1868:
taking values in the complex numbers. See for example
1474:{\displaystyle i\partial {\bar {\partial }}\psi >0}
1807:
1756:
1693:
1659:{\displaystyle \{z\mid -\infty \leq \psi (z)\leq c\}}
1616:
1588:
1560:
1540:
1487:
1443:
1419:
1399:
1366:
1346:
1326:
1300:
1280:
1193:
1173:
1153:
1114:
1076:
1010:
912:
739:
668:
574:
535:
515:
489:
469:
445:
281:
243:
219:
192:
169:
132:
112:
88:
1294:
is holomorphically spreadable, i.e. for every point
54:
dimensions. They were introduced by and named after
2033:"Stein spaces characterized by their endomorphisms"
1393:) exhaustive function, i.e. a smooth real function
1841:
1793:
1731:
1687:, the field of complex tangencies to the preimage
1658:
1594:
1566:
1546:
1526:
1473:
1425:
1405:
1372:
1352:
1332:
1312:
1286:
1217:
1179:
1159:
1129:
1091:
1046:
996:
891:
718:
607:
560:
521:
501:
475:
451:
423:
255:
225:
198:
178:
151:
118:
94:
2340:Zhang, Jing (2008). "Algebraic Stein varieties".
2037:Transactions of the American Mathematical Society
719:{\displaystyle H^{1}(X,{\mathcal {O}}_{X}^{*})=0}
2380:
1891:set of analogies, Stein manifolds correspond to
619:Non-compact Riemann surfaces are Stein manifolds
344:
1909:
2245:
2219:
1898:Stein manifolds are in some sense dual to the
1269:for Riemann surfaces, due to Behnke and Stein.
1992:, Handlebody construction of Stein surfaces,
1527:{\displaystyle \{z\in X\mid \psi (z)\leq c\}}
2151:(including a proof of the embedding theorem)
1914:Every compact smooth manifold of dimension 2
1683:such that, away from the critical points of
1653:
1617:
1521:
1488:
1244:Every Stein manifold of (complex) dimension
2274:
2097:
2010:International Mathematics Research Notices
1240:(because the embedding is biholomorphic).
1065:Properties and examples of Stein manifolds
233:is holomorphically convex, i.e. for every
2353:
2220:Grauert, Hans; Remmert, Reinhold (1979),
2170:
2124:
2058:
2048:
1949:
1918:, which has only handles of index ≤
1196:
1117:
1079:
1031:
843:
317:
311:
2100:Stein Manifolds and Holomorphic Mappings
1340:holomorphic functions defined on all of
2075:
2030:
1057:This is related to the solution of the
1047:{\displaystyle H^{2}(X,\mathbb {Z} )=0}
730:leads to the following exact sequence:
14:
2381:
1574:. This is a solution to the so-called
561:{\displaystyle f\in {\mathcal {O}}(X)}
483:is holomorphically separable, i.e. if
2339:
2303:
2154:
59:
1978:International Journal of Mathematics
1794:{\displaystyle f^{-1}(-\infty ,c).}
1218:{\displaystyle \mathbb {C} ^{2n+1}}
654:(1956), states moreover that every
24:
1842:{\displaystyle f^{-1}(-\infty ,c)}
1827:
1776:
1675:with a real-valued Morse function
1629:
1453:
1447:
974:
935:
875:
801:
762:
691:
544:
395:
384:
210:if the following conditions hold:
135:
25:
2400:
1922:, has a Stein structure provided
1610:. A Stein domain is the preimage
152:{\displaystyle {\mathcal {O}}(X)}
27:In mathematics, in the theory of
1130:{\displaystyle \mathbb {C} ^{n}}
1092:{\displaystyle \mathbb {C} ^{n}}
2060:10.1090/S0002-9947-2010-05104-9
1743:that induces an orientation on
1732:{\displaystyle X_{c}=f^{-1}(c)}
1999:
1983:
1967:
1943:
1836:
1821:
1785:
1770:
1726:
1720:
1644:
1638:
1512:
1506:
1456:
1433:(which can be assumed to be a
1387:strongly pseudoconvex manifold
1035:
1021:
985:
962:
946:
923:
886:
863:
850:
847:
833:
820:
817:
789:
776:
773:
750:
707:
679:
646:Another result, attributed to
639:and Stein (1948) asserts that
608:{\displaystyle f(x)\neq f(y).}
599:
593:
584:
578:
555:
549:
406:
400:
377:
373:
367:
360:
336:
332:
326:
319:
288:
146:
140:
13:
1:
2342:Mathematical Research Letters
2024:
77:
2081:Lectures on Riemann surfaces
1926: > 2, and when
1910:Relation to smooth manifolds
1602:invites a generalization of
1248:has the homotopy type of an
627:be a connected, non-compact
7:
2364:10.4310/MRL.2008.v15.n4.a16
1957:Encyclopedia of Mathematics
1070:The standard complex space
266:holomorphically convex hull
10:
2405:
2098:ForstneriÄŤ, Franc (2011).
1980:vol. 1, no 1 (1990) 29–46.
1932:Thurston–Bennequin framing
728:exponential sheaf sequence
256:{\displaystyle K\subset X}
2260:10.1007/s00208-009-0463-0
2108:10.1007/978-3-642-22250-4
1950:Onishchik, A.L. (2001) ,
1870:Cartan's theorems A and B
656:holomorphic vector bundle
29:several complex variables
2031:Andrist, Rafael (2010).
1937:
1481:, such that the subsets
1252:-dimensional CW-complex.
2012:(1998), no.7, 371–381.
502:{\displaystyle x\neq y}
74:in algebraic geometry.
2222:Theory of Stein spaces
1843:
1795:
1733:
1660:
1596:
1568:
1554:for every real number
1548:
1528:
1475:
1427:
1407:
1374:
1354:
1334:
1314:
1313:{\displaystyle x\in X}
1288:
1219:
1181:
1161:
1131:
1093:
1048:
998:
893:
720:
609:
562:
523:
503:
477:
453:
425:
257:
227:
200:
180:
153:
120:
96:
2277:Annals of Mathematics
2248:Mathematische Annalen
2158:Annals of Mathematics
1994:Annals of Mathematics
1878:analytic continuation
1866:holomorphic functions
1844:
1796:
1734:
1661:
1597:
1595:{\displaystyle \psi }
1582:(1911). The function
1569:
1549:
1529:
1476:
1428:
1408:
1406:{\displaystyle \psi }
1375:
1355:
1335:
1315:
1289:
1274:Every Stein manifold
1220:
1187:can be embedded into
1182:
1167:of complex dimension
1162:
1132:
1094:
1059:second Cousin problem
1049:
999:
894:
721:
643:is a Stein manifold.
610:
563:
524:
504:
478:
454:
426:
258:
228:
201:
181:
161:holomorphic functions
154:
121:
106:of complex dimension
97:
1996:148, (1998) 619–693.
1805:
1754:
1691:
1614:
1586:
1558:
1538:
1485:
1441:
1417:
1397:
1364:
1344:
1324:
1298:
1278:
1261:is a Stein manifold
1191:
1171:
1151:
1137:is a Stein manifold.
1112:
1106:domain of holomorphy
1099:is a Stein manifold.
1074:
1008:
910:
737:
666:
572:
533:
529:, then there exists
513:
487:
467:
443:
279:
241:
217:
190:
167:
130:
110:
86:
816:
706:
159:denote the ring of
2318:10.1007/bf02054949
1900:elliptic manifolds
1839:
1791:
1729:
1656:
1592:
1564:
1544:
1524:
1471:
1423:
1403:
1370:
1350:
1330:
1310:
1284:
1215:
1177:
1157:
1127:
1089:
1044:
994:
904:Cartan's theorem B
889:
798:
716:
688:
605:
558:
519:
509:are two points in
499:
473:
449:
421:
358:
253:
223:
196:
179:{\displaystyle X.}
176:
149:
116:
92:
2389:Complex manifolds
2161:, Second Series,
2139:978-0-444-88446-6
2117:978-3-642-22249-8
1882:analytic function
1741:contact structure
1567:{\displaystyle c}
1547:{\displaystyle X}
1459:
1426:{\displaystyle X}
1373:{\displaystyle x}
1353:{\displaystyle X}
1333:{\displaystyle n}
1287:{\displaystyle X}
1180:{\displaystyle n}
1160:{\displaystyle X}
522:{\displaystyle X}
476:{\displaystyle X}
452:{\displaystyle X}
383:
343:
291:
226:{\displaystyle X}
199:{\displaystyle X}
119:{\displaystyle n}
95:{\displaystyle X}
33:complex manifolds
16:(Redirected from
2396:
2375:
2357:
2336:
2300:
2271:
2242:
2215:
2174:
2150:
2121:
2093:
2072:
2062:
2052:
2043:(5): 2341–2355.
2019:
2003:
1997:
1987:
1981:
1974:Yakov Eliashberg
1971:
1965:
1964:
1947:
1893:affine varieties
1874:sheaf cohomology
1848:
1846:
1845:
1840:
1820:
1819:
1800:
1798:
1797:
1792:
1769:
1768:
1738:
1736:
1735:
1730:
1719:
1718:
1703:
1702:
1665:
1663:
1662:
1657:
1601:
1599:
1598:
1593:
1573:
1571:
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1565:
1553:
1551:
1550:
1545:
1533:
1531:
1530:
1525:
1480:
1478:
1477:
1472:
1461:
1460:
1452:
1432:
1430:
1429:
1424:
1412:
1410:
1409:
1404:
1391:plurisubharmonic
1379:
1377:
1376:
1371:
1359:
1357:
1356:
1351:
1339:
1337:
1336:
1331:
1319:
1317:
1316:
1311:
1293:
1291:
1290:
1285:
1224:
1222:
1221:
1216:
1214:
1213:
1199:
1186:
1184:
1183:
1178:
1166:
1164:
1163:
1158:
1136:
1134:
1133:
1128:
1126:
1125:
1120:
1098:
1096:
1095:
1090:
1088:
1087:
1082:
1053:
1051:
1050:
1045:
1034:
1020:
1019:
1003:
1001:
1000:
995:
984:
983:
978:
977:
961:
960:
945:
944:
939:
938:
922:
921:
898:
896:
895:
890:
885:
884:
879:
878:
862:
861:
846:
832:
831:
815:
810:
805:
804:
788:
787:
772:
771:
766:
765:
749:
748:
725:
723:
722:
717:
705:
700:
695:
694:
678:
677:
614:
612:
611:
606:
567:
565:
564:
559:
548:
547:
528:
526:
525:
520:
508:
506:
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500:
482:
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417:
413:
412:
409:
399:
398:
381:
380:
363:
357:
339:
322:
293:
292:
284:
263:, the so-called
262:
260:
259:
254:
232:
230:
229:
224:
205:
203:
202:
197:
185:
183:
182:
177:
158:
156:
155:
150:
139:
138:
125:
123:
122:
117:
104:complex manifold
101:
99:
98:
93:
68:affine varieties
21:
2404:
2403:
2399:
2398:
2397:
2395:
2394:
2393:
2379:
2378:
2289:10.2307/1970468
2232:
2140:
2126:Hörmander, Lars
2118:
2091:
2027:
2022:
2004:
2000:
1988:
1984:
1972:
1968:
1948:
1944:
1940:
1912:
1858:
1812:
1808:
1806:
1803:
1802:
1761:
1757:
1755:
1752:
1751:
1748:
1711:
1707:
1698:
1694:
1692:
1689:
1688:
1615:
1612:
1611:
1587:
1584:
1583:
1559:
1556:
1555:
1539:
1536:
1535:
1534:are compact in
1486:
1483:
1482:
1451:
1450:
1442:
1439:
1438:
1418:
1415:
1414:
1398:
1395:
1394:
1365:
1362:
1361:
1345:
1342:
1341:
1325:
1322:
1321:
1299:
1296:
1295:
1279:
1276:
1275:
1259:Riemann surface
1200:
1195:
1194:
1192:
1189:
1188:
1172:
1169:
1168:
1152:
1149:
1148:
1121:
1116:
1115:
1113:
1110:
1109:
1083:
1078:
1077:
1075:
1072:
1071:
1067:
1030:
1015:
1011:
1009:
1006:
1005:
979:
973:
972:
971:
956:
952:
940:
934:
933:
932:
917:
913:
911:
908:
907:
880:
874:
873:
872:
857:
853:
842:
827:
823:
811:
806:
800:
799:
783:
779:
767:
761:
760:
759:
744:
740:
738:
735:
734:
701:
696:
690:
689:
673:
669:
667:
664:
663:
637:Heinrich Behnke
629:Riemann surface
621:
573:
570:
569:
543:
542:
534:
531:
530:
514:
511:
510:
488:
485:
484:
468:
465:
464:
444:
441:
440:
394:
393:
376:
359:
347:
335:
318:
316:
312:
301:
297:
283:
282:
280:
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242:
239:
238:
218:
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214:
191:
188:
187:
168:
165:
164:
134:
133:
131:
128:
127:
111:
108:
107:
87:
84:
83:
80:
23:
22:
15:
12:
11:
5:
2402:
2392:
2391:
2377:
2376:
2348:(4): 801–814.
2337:
2301:
2272:
2243:
2230:
2217:
2181:10.2307/121005
2152:
2138:
2122:
2116:
2095:
2089:
2073:
2026:
2023:
2021:
2020:
2006:Selman Akbulut
1998:
1982:
1966:
1952:"Levi problem"
1941:
1939:
1936:
1911:
1908:
1872:, relating to
1862:
1861:
1856:
1838:
1835:
1832:
1829:
1826:
1823:
1818:
1815:
1811:
1790:
1787:
1784:
1781:
1778:
1775:
1772:
1767:
1764:
1760:
1746:
1728:
1725:
1722:
1717:
1714:
1710:
1706:
1701:
1697:
1668:
1667:
1655:
1652:
1649:
1646:
1643:
1640:
1637:
1634:
1631:
1628:
1625:
1622:
1619:
1604:Stein manifold
1591:
1578:, named after
1563:
1543:
1523:
1520:
1517:
1514:
1511:
1508:
1505:
1502:
1499:
1496:
1493:
1490:
1470:
1467:
1464:
1458:
1455:
1449:
1446:
1435:Morse function
1422:
1402:
1382:
1381:
1369:
1349:
1329:
1309:
1306:
1303:
1283:
1271:
1270:
1263:if and only if
1254:
1253:
1234:
1233:
1212:
1209:
1206:
1203:
1198:
1176:
1156:
1144:
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72:affine schemes
56:Karl Stein
37:Stein manifold
9:
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39:is a complex
38:
34:
30:
19:
2355:math/0610886
2345:
2341:
2309:
2305:
2283:(1): 34–46.
2280:
2276:
2251:
2247:
2221:
2172:math/9803019
2162:
2156:
2129:
2099:
2080:
2040:
2036:
2001:
1990:Robert Gompf
1985:
1969:
1955:
1945:
1927:
1923:
1919:
1915:
1913:
1897:
1886:
1863:
1854:
1744:
1684:
1680:
1676:
1672:
1607:
1603:
1580:Eugenio Levi
1576:Levi problem
1575:
1386:
1320:, there are
1249:
1245:
1235:
1056:
1004:, therefore
901:
659:
652:Helmut Röhrl
648:Hans Grauert
645:
640:
624:
622:
436:
264:
207:
81:
63:
48:
45:vector space
36:
26:
2312:: 201–222,
1849:is a Stein
906:shows that
64:Stein space
41:submanifold
18:Stein space
2306:Math. Ann.
2025:References
1230:proper map
568:such that
439:subset of
435:is also a
78:Definition
2334:122647212
2254:: 25–33.
2189:0003-486X
2050:0809.3919
1962:EMS Press
1828:∞
1825:−
1814:−
1801:That is,
1777:∞
1774:−
1763:−
1713:−
1648:≤
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1633:≤
1630:∞
1627:−
1624:∣
1590:ψ
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1504:ψ
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1305:∈
851:⟶
821:⟶
813:∗
777:⟶
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588:≠
540:∈
494:≠
391:∈
385:∀
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341:≤
306:∈
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248:⊂
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2268:10734808
2213:17709531
2128:(1990),
2079:(1981),
2069:14903691
186:We call
126:and let
82:Suppose
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2326:0043219
2297:1970468
2240:0580152
2205:1668563
2148:1045639
2017:1623402
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1851:filling
1437:) with
633:theorem
437:compact
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382:
2350:arXiv
2330:S2CID
2293:JSTOR
2264:S2CID
2209:S2CID
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2167:arXiv
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2045:arXiv
1938:Notes
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1225:by a
102:is a
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2226:ISBN
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