Pseudoconvexity and the problem of Levi

Siu, Yum-Tong

doi:10.1090/S0002-9904-1978-14483-8

Pseudoconvexity and the problem of Levi
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Bull. Amer. Math. Soc. 84 (1978), 481-512

References

Aldo Andreotti and Raghavan Narasimhan, Oka’s Heftungslemma and the Levi problem for complex spaces, Trans. Amer. Math. Soc. 111 (1964), 345–366. MR 159961, DOI 10.1090/S0002-9947-1964-0159961-3
Aldo Andreotti and Edoardo Vesentini, Carleman estimates for the Laplace-Beltrami equation on complex manifolds, Inst. Hautes Études Sci. Publ. Math. 25 (1965), 81–130. MR 175148, DOI 10.1007/BF02684398
Eric Bedford and John Erik Fornaess, A construction of peak functions on weakly pseudoconvex domains, Ann. of Math. (2) 107 (1978), no. 3, 555–568. MR 492400, DOI 10.2307/1971128
H. Behnke and K. Stein, Konvergente Folgen von Regularitätsbereichen und die Meromorphiekonvexität, Math. Ann. 116 (1939), no. 1, 204–216 (German). MR 1513225, DOI 10.1007/BF01597355
H. Behnke, Zur Theorie der Singularitäten der Funktionen mehrerer komplexen Veränderlichen, Math. Ann. 108 (1933), no. 1, 91–104 (German). MR 1512836, DOI 10.1007/BF01452824
Errett Bishop, Mappings of partially analytic spaces, Amer. J. Math. 83 (1961), 209–242. MR 123732, DOI 10.2307/2372953
Thomas Bloom and Ian Graham, A geometric characterization of points of type $m$ on real submanifolds of $\textbf {C}^{n}$, J. Differential Geometry 12 (1977), no. 2, 171–182. MR 492369
Hans J. Bremermann, Über die Äquivalenz der pseudokonvexen Gebiete und der Holomorphiegebiete im Raum von $n$ komplexen Veränderlichen, Math. Ann. 128 (1954), 63–91 (German). MR 71088, DOI 10.1007/BF01360125
Jérôme Brun, Sur le problème de Levi dans certains fibres, Manuscripta Math. 14 (1974), 217–222 (French, with English summary). MR 364686, DOI 10.1007/BF01171407
Jérôme Brun, Le problème de Levi dans les fibrés à base de Stein et à fibre une courbe compacte, Ann. Inst. Fourier (Grenoble) 27 (1977), no. 3, ix, 17–28 (French, with English summary). MR 486669
Lutz Bungart, On analytic fiber bundles. I. Holomorphic fiber bundles with infinite dimensional fibers, Topology 7 (1967), 55–68. MR 222338, DOI 10.1016/0040-9383(86)90015-7
James A. Carlson and Reese Harvey, A remark on the universal cover of a Moishezon space, Duke Math. J. 43 (1976), no. 3, 497–500. MR 412470
Henri Cartan and Peter Thullen, Zur Theorie der Singularitäten der Funktionen mehrerer komplexen Veränderlichen, Math. Ann. 106 (1932), no. 1, 617–647 (German). MR 1512777, DOI 10.1007/BF01455905
Jeff Cheeger and Detlef Gromoll, On the structure of complete manifolds of nonnegative curvature, Ann. of Math. (2) 96 (1972), 413–443. MR 309010, DOI 10.2307/1970819
Klas Diederich and John Erik Fornaess, Pseudoconvex domains: an example with nontrivial Nebenhülle, Math. Ann. 225 (1977), no. 3, 275–292. MR 430315, DOI 10.1007/BF01425243
Klas Diederich and John Erik Fornaess, Pseudoconvex domains: bounded strictly plurisubharmonic exhaustion functions, Invent. Math. 39 (1977), no. 2, 129–141. MR 437806, DOI 10.1007/BF01390105
Klas Diederich and John E. Fornaess, Pseudoconvex domains with real-analytic boundary, Ann. of Math. (2) 107 (1978), no. 2, 371–384. MR 477153, DOI 10.2307/1971120
Ferdinand Docquier and Hans Grauert, Levisches Problem und Rungescher Satz für Teilgebiete Steinscher Mannigfaltigkeiten, Math. Ann. 140 (1960), 94–123 (German). MR 148939, DOI 10.1007/BF01360084
Georges Elencwajg, Pseudo-convexité locale dans les variétés kahlériennes, Ann. Inst. Fourier (Grenoble) 25 (1975), no. 2, xv, 295–314 (French, with French summary). MR 387662
Gerd Fischer, Holomorph-vollständige Faserbündel, Math. Ann. 180 (1969), 341–348 (German). MR 241694, DOI 10.1007/BF01351887
Gerd Fischer, Fibrés holomorphes au-dessus d’un espace de Stein, Espaces Analytiques (Séminaire, Bucharest, 1969) Editura Acad. R.S.R., Bucharest, 1971, pp. 57–69 (French). MR 0285726
Gerd Fischer, Hilbert spaces of holomorphic functions on bounded domains, Manuscripta Math. 3 (1970), 305–314. MR 269879, DOI 10.1007/BF01338662
John Erik Fornaess, An increasing sequence of Stein manifolds whose limit is not Stein, Math. Ann. 223 (1976), no. 3, 275–277. MR 417448, DOI 10.1007/BF01360958
John Erik Fornaess, $2$ dimensional counterexamples to generalizations of the Levi problem, Math. Ann. 230 (1977), no. 2, 169–173. MR 486625, DOI 10.1007/BF01370661
John Erik Fornaess, Peak points on weakly pseudoconvex domains, Math. Ann. 227 (1977), no. 2, 173–175. MR 430316, DOI 10.1007/BF01350193
John Erik Fornaess and Edgar Lee Stout, Polydiscs in complex manifolds, Math. Ann. 227 (1977), no. 2, 145–153. MR 435441, DOI 10.1007/BF01350191
Reiko Fujita, Domaines sans point critique intérieur sur l’espace projectif complexe, J. Math. Soc. Japan 15 (1963), 443–473 (French). MR 159034, DOI 10.2969/jmsj/01540443
Hans Grauert, Charakterisierung der Holomorphiegebiete durch die vollständige Kählersche Metrik, Math. Ann. 131 (1956), 38–75 (German). MR 77651, DOI 10.1007/BF01354665
Hans Grauert, On Levi’s problem and the imbedding of real-analytic manifolds, Ann. of Math. (2) 68 (1958), 460–472. MR 98847, DOI 10.2307/1970257
Hans Grauert, Analytische Faserungen über holomorph-vollständigen Räumen, Math. Ann. 135 (1958), 263–273 (German). MR 98199, DOI 10.1007/BF01351803
Hans Grauert, Über Modifikationen und exzeptionelle analytische Mengen, Math. Ann. 146 (1962), 331–368 (German). MR 137127, DOI 10.1007/BF01441136
Hans Grauert, Bemerkenswerte pseudokonvexe Mannigfaltigkeiten, Math. Z. 81 (1963), 377–391 (German). MR 168798, DOI 10.1007/BF01111528
Hans Grauert and Reinhold Remmert, Konvexität in der komplexen Analysis. Nicht-holomorph-konvexe Holomorphiegebiete und Anwendungen auf die Abbildungstheorie, Comment. Math. Helv. 31 (1956), 152–160, 161–183 (German). MR 88028, DOI 10.1007/BF02564357
Hans Grauert and Reinhold Remmert, Singularitäten komplexer Mannigfaltigkeiten und Riemannsche Gebiete, Math. Z. 67 (1957), 103–128 (German). MR 87186, DOI 10.1007/BF01258848
Hans Grauert and Oswald Riemenschneider, Verschwindungssätze für analytische Kohomologiegruppen auf komplexen Räumen, Invent. Math. 11 (1970), 263–292 (German). MR 302938, DOI 10.1007/BF01403182
R. E. Greene and H. Wu, Curvature and complex analysis, Bull. Amer. Math. Soc. 77 (1971), 1045–1049. MR 283240, DOI 10.1090/S0002-9904-1971-12856-2
R. E. Greene and H. Wu, Some function-theoretic properties of noncompact Kähler manifolds, Differential geometry (Proc. Sympos. Pure Math., Vol. XXVII, Part 2, Stanford Univ., Stanford, Calif., 1973) Amer. Math. Soc., Providence, R.I., 1975, pp. 33–41. MR 0379898
R. E. Greene and H. Wu, $C^{\infty }$ convex functions and manifolds of positive curvature, Acta Math. 137 (1976), no. 3-4, 209–245. MR 458336, DOI 10.1007/BF02392418
R. E. Greene and H. Wu, Analysis on noncompact Kähler manifolds, Several complex variables (Proc. Sympos. Pure Math., Vol. XXX, Part 2, Williams Coll., Williamstown, Mass., 1975) Amer. Math. Soc., Providence, R.I., 1977, pp. 69–100. MR 0460699
Phillip A. Griffiths, Two theorems on extensions of holomorphic mappings, Invent. Math. 14 (1971), 27–62. MR 293123, DOI 10.1007/BF01418742
Monique Hakim and Nessim Sibony, Quelques conditions pour l’existence de fonctions pics dans des domaines pseudoconvexes, Duke Math. J. 44 (1977), no. 2, 399–406 (French). MR 457784
Fritz Hartogs, Zur Theorie der analytischen Funktionen mehrerer unabhängiger Veränderlichen, insbesondere über die Darstellung derselben durch Reihen, welche nach Potenzen einer Veränderlichen fortschreiten, Math. Ann. 62 (1906), no. 1, 1–88 (German). MR 1511365, DOI 10.1007/BF01448415
André Hirschowitz, Corrections: “Sur certains fibrés holomorphes à base et fibre de Stein” (C. R. Acad. Sci. Paris Sér. A-B 276 (1973), A925–A927), C. R. Acad. Sci. Paris Sér. A 278 (1974), 89–91 (French). MR 355121
André Hirschowitz, Pseudoconvexité au-dessus d’espaces plus ou moins homogènes, Invent. Math. 26 (1974), 303–322 (French). MR 357857, DOI 10.1007/BF01425555
A. Hirschowitz, Domaines de Stein et fonctions holomorphes bornés, Math. Ann. 213 (1975), 185–193 (French). MR 393563, DOI 10.1007/BF01343951
André Hirschowitz, Le problème de Lévi pour les espaces homogènes, Bull. Soc. Math. France 103 (1975), no. 2, 191–201. MR 399510, DOI 10.24033/bsmf.1801
Lars Hörmander, $L^{2}$ estimates and existence theorems for the $\bar \partial$ operator, Acta Math. 113 (1965), 89–152. MR 179443, DOI 10.1007/BF02391775
C. O. Kiselman, On entire functions of exponential type and indicators of analytic functionals, Acta Math. 117 (1967), 1–35. MR 210940, DOI 10.1007/BF02395038
J. J. Kohn, Boundary behavior of $\delta$ on weakly pseudo-convex manifolds of dimension two, J. Differential Geometry 6 (1972), 523–542. MR 322365
J. J. Kohn and L. Nirenberg, A pseudo-convex domain not admitting a holomorphic support function, Math. Ann. 201 (1973), 265–268. MR 330513, DOI 10.1007/BF01428194
Konrad Königsberger, Über die Holomorphie-Vollständigkeit lokal trivialer Faserräume, Math. Ann. 189 (1970), 178–184 (German). MR 268410, DOI 10.1007/BF01352443
P. Le Barz, À propos des revêtements ramifiés d’espaces de Stein, Math. Ann. 222 (1976), no. 1, 63–69 (French). MR 417449, DOI 10.1007/BF01418244
Pierre Lelong, Fonctionnelles analytiques et fonctions entières ($n$ variables), Séminaire de Mathématiques Supérieures, No. 13 (Été, vol. 1967, Les Presses de l’Université de Montréal, Montreal, Que., 1968 (French). MR 0466606
Bernard Malgrange, La cohomologie d’une variété analytique complexe à bord pseudo-convexe n’est pas nécessairement séparée, C. R. Acad. Sci. Paris Sér. A-B 280 (1975), Aii, A93–A95 (French, with English summary). MR 372241
Andrew Markoe, Runge families and increasing unions of Stein spaces, Bull. Amer. Math. Soc. 82 (1976), no. 5, 787–788. MR 414934, DOI 10.1090/S0002-9904-1976-14166-3
Yasuo Matsugu, The Levi problem for a product manifold, Pacific J. Math. 46 (1973), 231–233. MR 322207, DOI 10.2140/pjm.1973.46.231
Yozô Matsushima and Akihiko Morimoto, Sur certains espaces fibrés holomorphes sur une variété de Stein, Bull. Soc. Math. France 88 (1960), 137–155 (French). MR 123739, DOI 10.24033/bsmf.1548
Daniel Michel, Sur les ouverts pseudo-convexes des espaces homogènes, C. R. Acad. Sci. Paris Sér. A-B 283 (1976), no. 10, Aiii, A779–A782 (French, with English summary). MR 460726
A. Morimoto, Non-compact complex Lie groups without non-constant holomorphic functions, Proc. Conf. Complex Analysis (Minneapolis, 1964) Springer, Berlin, 1965, pp. 256–272. MR 0181702
Raghavan Narasimhan, Imbedding of holomorphically complete complex spaces, Amer. J. Math. 82 (1960), 917–934. MR 148942, DOI 10.2307/2372949
Raghavan Narasimhan, The Levi problem for complex spaces, Math. Ann. 142 (1960/61), 355–365. MR 148943, DOI 10.1007/BF01451029
Raghavan Narasimhan, The Levi problem in the theory of functions of several complex variables, Proc. Internat. Congr. Mathematicians (Stockholm, 1962) Inst. Mittag-Leffler, Djursholm, 1963, pp. 385–388. MR 0176096
François Norguet, Sur les domaines d’holomorphie des fonctions uniformes de plusieurs variables complexes. (Passage du local au global.), Bull. Soc. Math. France 82 (1954), 137–159 (French). MR 71087, DOI 10.24033/bsmf.1448
Kiyosi Oka, Sur les fonctions analytiques de plusieurs variables. VI. Domaines pseudoconvexes, Tôhoku Math. J. 49 (1942), 15–52 (French). MR 14470
Kiyoshi Oka, Sur les fonctions analytiques de plusieurs variables. IX. Domaines finis sans point critique intérieur, Jpn. J. Math. 23 (1953), 97–155 (1954) (French). MR 71089, DOI 10.4099/jjm1924.23.0_{9}7
Peter Pflug, Quadratintegrable holomorphe Funktionen und die Serre-Vermutung, Math. Ann. 216 (1975), no. 3, 285–288 (German). MR 382717, DOI 10.1007/BF01430969
Peter Pflug, Glatte Holomorphiegebiete mit plurisubharmonischer innerer Randfunktion sind Banach-Stein, Ark. Mat. 14 (1976), no. 1, 55–58. MR 427668, DOI 10.1007/BF02385823
Reinhold Remmert, Sur les espaces analytiques holomorphiquement séparables et holomorphiquement convexes, C. R. Acad. Sci. Paris 243 (1956), 118–121 (French). MR 79808
Rolf Richberg, Stetige streng pseudokonvexe Funktionen, Math. Ann. 175 (1968), 257–286 (German). MR 222334, DOI 10.1007/BF02063212
Jean-Pierre Serre, Quelques problèmes globaux relatifs aux variétés de Stein, Colloque sur les fonctions de plusieurs variables, tenu à Bruxelles, 1953, Georges Thone, Liège; Masson & Cie, Paris, 1953, pp. 57–68 (French). MR 0064155
Bernard Shiffman, Extension of holomorphic maps into hermitian manifolds, Math. Ann. 194 (1971), 249–258. MR 291507, DOI 10.1007/BF01350128
Nessim Sibony, Fibrés holomorphes et métrique de Carathéodory, C. R. Acad. Sci. Paris Sér. A 279 (1974), 261–264 (French). MR 352550
Yum Tong Siu, All plane domains are Banach-Stein, Manuscripta Math. 14 (1974), 101–105. MR 361157, DOI 10.1007/BF01637626
Yum Tong Siu, Holomorphic fiber bundles whose fibers are bounded Stein domains with zero first Betti number, Math. Ann. 219 (1976), no. 2, 171–192. MR 390303, DOI 10.1007/BF01351901
Yum Tong Siu, The Levi problem, Several complex variables (Proc. Sympos. Pure Math., Vol. XXX, Part 2, Williams Coll., Williamstown, Mass., 1975) Amer. Math. Soc., Providence, R.I., 1977, pp. 45–48. MR 0473242
Henri Skoda, Fibrés holomorphes à base et à fibre de Stein, C. R. Acad. Sci. Paris Sér. A-B 284 (1977), no. 19, A1199–A1202. MR 437802
H. Skoda, Fibrés holomorphes à base et à fibre de Stein, Invent. Math. 43 (1977), no. 2, 97–107 (French). MR 508091, DOI 10.1007/BF01390000
Karl Stein, Überlagerungen holomorph-vollständiger komplexer Räume, Arch. Math. (Basel) 7 (1956), 354–361 (German). MR 84836, DOI 10.1007/BF01900686
Jean-Luc Stehlé, Fonctions plurisousharmoniques et convexité holomorphe de certains fibrés analytiques, C. R. Acad. Sci. Paris Sér. A 279 (1974), 235–238 (French). MR 374493
Osamu Suzuki, Pseudoconvex domains on a Kähler manifold with positive holomorphic bisectional curvature, Publ. Res. Inst. Math. Sci. 12 (1976/77), no. 1, 191–214. MR 0450612, DOI 10.2977/prims/1195190963
Akira Takeuchi, Domaines pseudoconvexes infinis et la métrique riemannienne dans un espace projectif, J. Math. Soc. Japan 16 (1964), 159–181 (French). MR 173789, DOI 10.2969/jmsj/01620109
Akira Takeuchi, Domaines pseudoconvexes sur les variétés kählériennes, J. Math. Kyoto Univ. 6 (1967), 323–357 (French). MR 217335, DOI 10.1215/kjm/1250524335