Generically strongly q-convex complex manifolds  [ Variétés complexes génériquement fortement q-convexes ]
Annales de l'Institut Fourier, Tome 51 (2001) no. 6, p. 1553-1598
On suppose que ϕ est une fonction analytique-réelle plurisousharmonique sur une variété complexe connexe et non-compacte X. Le résultat principal démontre que si l’ensemble analytique-réel des points où ϕ n’est pas fortement q-convexe est de dimension 2q+1 ou moins, alors presque tous les sous-niveaux assez grands de ϕ sont des variétés complexes fortement q-convexes. Pour X de dimension 2, c’est un cas spécial d’un théorème de Diederich et Ohsawa. Nous obtenons aussi une version de ce résultat dans le cas où ϕ est analytique réelle avec coins.
Suppose ϕ is a real analytic plurisubharmonic exhaustion function on a connected noncompact complex manifold X. The main result is that if the real analytic set of points at which ϕ is not strongly q-convex is of dimension at most 2q+1, then almost every sufficiently large sublevel of ϕ is strongly q-convex as a complex manifold. For X of dimension 2, this is a special case of a theorem of Diederich and Ohsawa. A version for ϕ real analytic with corners is also obtained.
DOI : https://doi.org/10.5802/aif.1866
Classification:  32E40,  32F10
Mots clés: cycles analytiques, convexe holomorphiquement, q complet
@article{AIF_2001__51_6_1553_0,
     author = {Napier, Terrence and Ramachandran, Mohan},
     title = {Generically strongly $q$-convex complex manifolds},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {51},
     number = {6},
     year = {2001},
     pages = {1553-1598},
     doi = {10.5802/aif.1866},
     zbl = {0996.32004},
     mrnumber = {1870640},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2001__51_6_1553_0}
}
Napier, Terrence; Ramachandran, Mohan. Generically strongly $q$-convex complex manifolds. Annales de l'Institut Fourier, Tome 51 (2001) no. 6, pp. 1553-1598. doi : 10.5802/aif.1866. https://www.numdam.org/item/AIF_2001__51_6_1553_0/

[Ba1] D. Barlet Espace analytique réduit des cycles analytiques complexes compacts d'un espace analytique complexe de dimension finie, Séminaire F. Norguet : Fonctions de plusieurs variables complexes 1974/75, Springer, Berlin-Heidelberg-New York (Lecture Notes in Math.) Tome vol. 482 (1975), pp. 1-158 | Zbl 0331.32008

[Ba2] D. Barlet Convexité de l'espace des cycles, Bull. Soc. Math. France, Tome 106 (1978), pp. 373-397 | Numdam | MR 518045 | Zbl 0395.32009

[Bi] E. Bishop Conditions for the analyticity of certain sets, Michigan Math. J., Tome 11 (1964), pp. 289-304 | Article | MR 168801 | Zbl 0143.30302

[BrC1] F. Bruhat; H. Cartan Sur la structure des sous-ensembles analytiques-réels, C. R. Acad. Sci. Paris, Tome 244 (1957), pp. 988-990 | MR 86108 | Zbl 0081.17201

[BrC2] F. Bruhat; H. Cartan Sur les composantes irréductibles d'un sous-ensemble analytique-réel, C. R. Acad. Sci. Paris, Tome 244 (1957), pp. 1123-1126 | MR 88528 | Zbl 0081.39101

[BrW] F. Bruhat; H. Whitney Quelque propriétés fondamentales des ensembles analytiques-réels, Comm. Math. Helv., Tome 33 (1959), pp. 132-160 | Article | MR 102094 | Zbl 0100.08101

[Cam] F. Campana Remarques sur le revêtement universel des variétés kählériennes compactes, Bull. Soc. Math. France, Tome 122 (1994) no. 2, pp. 255-284 | Numdam | MR 1273904 | Zbl 0810.32013

[Car] H. Cartan Quotients of complex analytic spaces, Contributions to function theory, Internat. colloq. function theory, Tata Inst. of Fundamental Research, Bombay (1960) | Zbl 0122.08702

[Co] M. Coltoiu Complete locally pluripolar sets, J. reine and angew. Math., Tome 412 (1990), pp. 108-112 | Article | MR 1074376 | Zbl 0711.32008

[De1] J.-P. Demailly Estimations L 2 pour l'opérateur ¯ d'un fibré vectoriel holomorphe semi-positif au-dessus d'une variété kählérienne complète, Ann. Sci. Ecole Norm. Sup., Tome 15 (1982), pp. 457-511 | Numdam | MR 690650 | Zbl 0507.32021

[De2] J.-P. Demailly Cohomology of q-convex spaces in top degrees, Math. Z., Tome 204 (1990), pp. 283-295 | Article | MR 1055992 | Zbl 0682.32017

[DiF1] K. Diederich; J. E. Fornaess Pseudoconvex domains: bounded strictly plurisubharmonic exhaustion functions, Invent. Math., Tome 39 (1977), pp. 129-141 | Article | MR 437806 | Zbl 0353.32025

[DiF2] K. Diederich; J. E. Fornaess Pseudoconvex domains: existence of Stein neighborhoods, Duke Math. J., Tome 44 (1977), pp. 641-662 | Article | MR 447634 | Zbl 0381.32014

[DiF3] K. Diederich; J. E. Fornaess Pseudoconvex domains with real-analytic boundary, Ann. Math., Tome 107 (1978), pp. 371-384 | Article | MR 477153 | Zbl 0378.32014

[DiO] K. Diederich; T. Ohsawa A Levi problem on two-dimensional complex manifolds, Math. Ann., Tome 261 (1982), pp. 255-261 | Article | MR 675738 | Zbl 0502.32010

[DoG] F. Docquier; H. Grauert Levisches Problem und Rungescher Satz für Teilgebiete Steinscher Mannigfaltigkeiten, Math. Ann., Tome 140 (1960), pp. 94-123 | Article | MR 148939 | Zbl 0095.28004

[Fr] M. Freeman Local complex foliation of real submanifolds, Math. Ann., Tome 209 (1974), pp. 1-30 | Article | MR 346185 | Zbl 0267.32006

[Fu] A. Fujiki Closedness of the Douady spaces of compact Kähler spaces, Publ. Res. Inst. Math. Sci., Tome 14 (1978/79) no. 1, pp. 1-52 | Article | MR 486648 | Zbl 0409.32016

[G] H. Grauert On Levi's problem and the imbedding of real analytic manifolds, Ann. Math., Tome 68 (1958), pp. 460-472 | Article | MR 98847 | Zbl 0108.07804

[GR] H. Grauert; O. Riemeschneider Kählersche Mannigfältigkeiten mit hyper-q-konvexen Rand, Problems in analysis (A Symposium in Honor of S. Bochner, Princeton 1969), Princeton University Press, Princeton (1970), pp. 61-79 | Zbl 0211.10302

[GW] R. Greene; H. Wu Embedding of open Riemannian manifolds by harmonic functions, Ann. Inst. Fourier (Grenoble), Tome 25 (1975) no. 1, pp. 215-235 | Article | Numdam | MR 382701 | Zbl 0307.31003

[HM] L.R. Hunt; J.J. Murray Plurisubharmonic functions and a generalized Dirichlet problem, Mich. Math. J, Tome 25 (1978), pp. 299-316 | Article | MR 512901 | Zbl 0378.32013

[Hu] A. Huckleberry The Levi problem on pseudoconvex manifolds which are not strongly pseudoconvex, Math. Ann., Tome 219 (1976), pp. 127-137 | Article | MR 409892 | Zbl 0313.32017

[L] T. Levi-Civita Sulle funzione di due o più variabli complesse, Rend. Accad. Naz. Lincei. V, Tome 14 (1905), pp. 492-499

[Na] S. Nakano Vanishing theorems for weakly 1-complete manifolds II, Publ. R.I.M.S., Kyoto, Tome 10 (1974), pp. 101-110 | Article | MR 382735 | Zbl 0298.32019

[NR1] T. Napier; M. Ramachandran Structure theorems for complete Kähler manifolds and applications to Lefschetz type theorems, Geom. Funct. Anal., Tome 5 (1995), pp. 809-851 | Article | MR 1354291 | Zbl 0860.53045

[NR2] T. Napier; M. Ramachandran The Bochner-Hartogs dichotomy for weakly 1-complete Kähler manifolds, Ann. Inst. Fourier (Grenoble), Tome 47 (1997) no. 5, pp. 1345-1365 | Article | Numdam | MR 1600387 | Zbl 0904.32008

[Ns] R. Narasimhan The Levi problem for complex spaces II, Math. Ann., Tome 146 (1962), pp. 195-216 | Article | MR 182747 | Zbl 0131.30801

[O] T. Ohsawa Completeness of noncompact analytic spaces, Publ. R.I.M.S., Kyoto, Tome 20 (1984), pp. 683-692 | Article | MR 759689 | Zbl 0568.32008

[Re] R. Remmert Reduction of complex spaces, Seminars on analytic functions, Inst. for Advanced Study, Princeton (1957) | Zbl 0095.06204

[Ri] R. Richberg Stetige streng pseudokonvexe Funktionen, Math. Ann., Tome 175 (1968), pp. 257-286 | Article | MR 222334 | Zbl 0153.15401

[Si1] Y.-T. Siu Every Stein subvariety admits a Stein neighborhood, Invent. Math., Tome 38 (1976), pp. 89-100 | Article | MR 435447 | Zbl 0343.32014

[Si2] Y.-T. Siu Complex-analyticity of harmonic maps, vanishing and Lefschetz theorems, J. Differential Geom., Tome 17 (1982), pp. 55-138 | MR 658472 | Zbl 0497.32025

[Ste] J.-L. Stehlé Fonctions plurisousharmoniques et convexité holomorphe de certains fibrés analytiques. Séminaire Pierre Lelong (Analyse), Séminaire Pierre Lelong (Analyse), Année 1973--1974, Springer, Berlin-Heidelberg-New York (Lect. Notes in Math.) Tome vol. 474 (1975), pp. 155-179 | Zbl 0309.32011

[Sto] W. Stoll The fiber integral is constant, Math. Zeitsch., Tome 104 (1968), pp. 65-73 | Article | MR 224868 | Zbl 0164.09302

[Wu] H. Wu On certain Kähler manifolds which are q-complete, Complex analysis of Several Variables, Amer. Math. Soc., Providence (Proceedings of Symposia in Pure Mathematics) Tome vol. 41 (1984), pp. 253-276 | Zbl 0552.32015