On the local pseudoconvexity of certain analytic families of
Annales de l'Institut Fourier, Volume 68 (2018) no. 7, p. 2811-2818

For a class of weakly 1-complete bundles over compact Riemann surfaces, for which canonical plurisubharmonic exhaustion functions on the total spaces are known, some cases are described where such functions can be extended to a plurisubharmonic exhaustion function on analytic families of the bundles. The nonextendable cases are also discussed.

Nous donnons des conditions pour que certaines fonctions analytiques plurisousharmoniques exhaustives sur des variétés faiblement 1-complètes qui sont des fibrés en droites affines au dessus de surfaces de Riemann soient extensibles à des familles analytiques de fonctions plurisousharmoniques exhaustives. Un exemple de famille non-extensible est également présenté.

Published online : 2019-05-24
DOI : https://doi.org/10.5802/aif.3226
Classification:  32E40,  32T05
Keywords: plurisubharmonic functions, pseudoconvexity
@article{AIF_2018__68_7_2811_0,
     author = {Ohsawa, Takeo},
     title = {On the local pseudoconvexity of certain analytic families of $\protect \mathbb{C}$},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {68},
     number = {7},
     year = {2018},
     pages = {2811-2818},
     doi = {10.5802/aif.3226},
     language = {en},
     url = {https://aif.centre-mersenne.org/item/AIF_2018__68_7_2811_0}
}
On the local pseudoconvexity of certain analytic families of $\protect \mathbb{C}$. Annales de l'Institut Fourier, Volume 68 (2018) no. 7, pp. 2811-2818. doi : 10.5802/aif.3226. https://aif.centre-mersenne.org/item/AIF_2018__68_7_2811_0/

[1] Andreotti, Aldo; Grauert, Hans Théorème de finitude pour la cohomologie des espaces complexes, Bull. Soc. Math. Fr., Tome 90 (1962), pp. 193-259 | Zbl 0106.05501

[2] Andreotti, Aldo; Narasimhan, Raghavan Oka’s Heftungslemma and the Levi problem for complex spaces, Trans. Am. Math. Soc., Tome 111 (1964), pp. 345-366 | Zbl 0134.06001

[3] Behnke, Heinrich; Stein, Karl Entwicklung analytischer Funktionen auf Riemannschen Flächen, Math. Ann., Tome 120 (1949), pp. 430-461 | Zbl 0038.23502

[4] Cartan, Henri Sur quelques progrès dans la théorie des fonctions analytiques de variables complexes entre 1930 et 1950, Miscellanea mathematica, Springer (1991), pp. 45-58 | Zbl 0733.32001

[5] Demailly, Jean-Pierre Un exemple de fibré holomorphe non de Stein à fibre 2 ayant pour base le disque ou le plan, Invent. Math., Tome 48 (1978), pp. 293-302 | Zbl 0372.32012

[6] Diederich, Klas; Fornaess, John E. A smooth pseudoconvex domain without pseudoconvex exhaustion, Manuscr. Math., Tome 39 (1982), pp. 119-123 | Zbl 0524.32010

[7] Diederich, Klas; Ohsawa, Takeo A Levi problem on two-dimensional complex manifolds, Math. Ann., Tome 261 (1982), pp. 255-261 | Zbl 0502.32010

[8] Diederich, Klas; Ohsawa, Takeo Harmonic mappings and disc bundles over compact Kähler manifolds, Publ. Res. Inst. Math. Sci., Tome 21 (1985), pp. 819-833 | Zbl 0601.32023

[9] Fornaess, John E. 2 dimensional counterexamples to generalizations of the Levi problem, Math. Ann., Tome 230 (1977), pp. 169-173 | Zbl 0379.32016

[10] Fornaess, John E. A counterexample for the Levi problem for branched Riemann domains over n , Math. Ann., Tome 234 (1978), pp. 275-277 | Zbl 0427.32014

[11] Fujita, Reiko Domaines sans point critique intérieur sur l’espace projectif complexe, J. Math. Soc. Japan, Tome 15 (1963), pp. 443-473 | Zbl 0138.06404

[12] Grauert, Hans On Levi’s problem and the imbedding of real-analytic manifolds, Ann. Math., Tome 68 (1958), pp. 460-472 | Zbl 0108.07804

[13] Grauert, Hans Bemerkenswerte pseudokonvexe Mannigfaltigkeiten, Math. Z., Tome 81 (1963), pp. 377-391 | Zbl 0151.09702

[14] Grauert, Hans The methods of the theory of functions of several complex variables, Miscellanea mathematica, Springer (1991), pp. 129-143 | Zbl 0737.32001

[15] Greene, Robert E.; Wu, Hung-Hsi Embedding of open Riemannian manifolds by harmonic functions, Ann. Inst. Fourier, Tome 25 (1975) no. 1, pp. 215-235 | Zbl 0307.31003

[16] Miebach, Christian Quotients of bounded homogeneous domains by cyclic groups, Osaka J. Math., Tome 47 (2010) no. 2, pp. 331-352 | Zbl 1197.32007

[17] Nakano, Shigeo Vanishing theorems for weakly 1-complete manifolds, Number theory, algebraic geometry and commutative algebra in honor of Yasuo Akizuki, Kinokuniya Book-Store Co. (1973), pp. 169-179 | Zbl 0272.14005

[18] Nemirovski, Stefan Stein domains with Levi-flat boundaries on compact complex surfaces, Mat. Zametki, Tome 66 (1999), pp. 632-635 (translation in Math. Notes 66 (1999), no. 3-4, p. 522-525)

[19] Ohsawa, Takeo A Stein domain with smooth boundary which has a product structure, Publ. Res. Inst. Math. Sci., Tome 18 (1982) no. 3, p. 1185-1186 | Zbl 0509.32016

[20] Ohsawa, Takeo Completeness of noncompact analytic spaces, Publ. Res. Inst. Math. Sci., Tome 20 (1984), pp. 683-692 | Zbl 0568.32008

[21] Ohsawa, Takeo On the extension of L 2 holomorphic functions. V: Effect of generalization, Nagoya Math. J., Tome 161 (2001), pp. 1-21 (erratum in ibid. 163 (2001), p. 229) | Zbl 0986.32002

[22] Ohsawa, Takeo Stability of pseudoconvexity of disc bundles over compact Riemann surfaces and application to a family of Galois coverings, Int. J. Math., Tome 26 (2015) no. 4, 1540003, 7 pages (Art. ID 1540003, 7 p.) | Article | Zbl 1325.32009

[23] Oka, Kiyosi Sur les fonctions analytiques de plusieurs variables VI. Domaines pseudoconvexes, Tôhoku Math. J., Tome 49 (1942), pp. 15-52 | Zbl 0060.24006

[24] Oka, Kiyosi Sur les fonctions analytiques de plusieurs variables IX. Domaines finis sans point critique intérieur, Jap. J. Math., Tome 23 (1953), pp. 97-155 | Zbl 0053.24302

[25] Skoda, Henri Fibrés holomorphes à base et à fibre de Stein, Invent. Math., Tome 43 (1977), pp. 97-107 | Zbl 0365.32018

[26] Takayama, Shigeharu Adjoint linear series on weakly 1-complete Kähler manifolds. I. Global projective embedding, Math. Ann., Tome 311 (1998) no. 3, pp. 501-531 | Zbl 0912.32021

[27] Takayama, Shigeharu The Levi problem and the structure theorem for non-negatively curved complete Kähler manifolds, J. Reine Angew. Math., Tome 504 (1998), pp. 139-157 | Zbl 0911.32022

[28] Takeuchi, A. Domaines pseudoconvexes infinis et la métrique riemannienne dans un espace projectif, J. Math. Soc. Japan, Tome 16 (1964), pp. 159-181 | Zbl 0141.08804

[29] Ueda, Tetsuo Modifications continues des variétés de Stein, Publ. Res. Inst. Math. Sci., Tome 13 (1977), pp. 681-686 | Zbl 0379.32014

[30] Ueda, Tetsuo Pseudoconvex domains over Grassmann manifolds, J. Math. Kyoto Univ., Tome 20 (1980), pp. 391-394 | Zbl 0456.32011

[31] Ueda, Tetsuo On the neighborhood of a compact complex curve with topologically trivial normal bundle, J. Math. Kyoto Univ., Tome 22 (1982), pp. 583-607 | Zbl 0519.32019