no. 7
On the local pseudoconvexity of certain analytic families of ℂ
[Sur la pseudoconvexité locale de certaines familles analytiques de ℂ]
Ohsawa, Takeo1
1 Graduate School of Mathematics Nagoya University 464-8602 Chikusaku Furocho Nagoya (Japan)
Annales de l'Institut Fourier, Tome 68 (2018) no. 7, pp. 2811-2818.

For a class of weakly 1-complete ℂ 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 ℂ 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.

Publié le :
DOI : 10.5802/aif.3226
Classification : 32E40, 32T05
Keywords: plurisubharmonic functions, pseudoconvexity
Mots-clés : fonctions plurisousharmoniques, pseudoconvexité

Ohsawa, Takeo 1

1 Graduate School of Mathematics Nagoya University 464-8602 Chikusaku Furocho Nagoya (Japan)
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
@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},
     pages = {2811--2818},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {68},
     number = {7},
     year = {2018},
     doi = {10.5802/aif.3226},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3226/}
}
TY  - JOUR
AU  - Ohsawa, Takeo
TI  - On the local pseudoconvexity of certain analytic families of $\protect \mathbb{C}$
JO  - Annales de l'Institut Fourier
PY  - 2018
SP  - 2811
EP  - 2818
VL  - 68
IS  - 7
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.3226/
DO  - 10.5802/aif.3226
LA  - en
ID  - AIF_2018__68_7_2811_0
ER  - 
%0 Journal Article
%A Ohsawa, Takeo
%T On the local pseudoconvexity of certain analytic families of $\protect \mathbb{C}$
%J Annales de l'Institut Fourier
%D 2018
%P 2811-2818
%V 68
%N 7
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.3226/
%R 10.5802/aif.3226
%G en
%F AIF_2018__68_7_2811_0
Ohsawa, Takeo. On the local pseudoconvexity of certain analytic families of $\protect \mathbb{C}$. Annales de l'Institut Fourier, Tome 68 (2018) no. 7, pp. 2811-2818. doi : 10.5802/aif.3226. https://aif.centre-mersenne.org/articles/10.5802/aif.3226/

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

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

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

[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

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

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

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

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

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

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

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

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

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

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

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

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

[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

[18] Nemirovski, Stefan Stein domains with Levi-flat boundaries on compact complex surfaces, Mat. Zametki, Volume 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., Volume 18 (1982) no. 3, pp. 1185-1186 | Zbl

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

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

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

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

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

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

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

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

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

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

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

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

Cité par Sources :