Levi's Problem for Pseudoconvex Homogeneous Manifolds

Gilligan, Bruce

Canadian mathematical bulletin, Tome 60 (2017) no. 4, pp. 736-746

Résumé

Suppose $G$ is a connected complex Lie group and $H$ is a closed complex subgroup. Then there exists a closed complex subgroup $J$ of $G$ containing $H$ such that the fibration $\pi :G/H\to $ $G/J$ is the holomorphic reduction of $G/H$ i.e., $G/J$ is holomorphically separable and $\mathcal{O}(G/H)\cong $ ${{\pi }^{*}}\mathcal{O}(G/J)$ . In this paper we prove that if $G/H$ is pseudoconvex, i.e., if $G/H$ admits a continuous plurisubharmonic exhaustion function, then $G/J$ is Stein and $J/H$ has no non-constant holomorphic functions.

Détail

DOI : 10.4153/CMB-2017-007-x

Mots-clés : 32M10, 32U10, 32A10, 32Q28, complex homogeneous manifold, plurisubharmonic exhaustion function, holomorphic reduction, Stein manifold, Remmert reduction, Hirschowitz annihilator

Bibliographie

[1] [1] Akhiezer, D. N., Lie group actions in complex analysis. Aspects of Mathematics, E27, Friedr. Vieweg & Sohn, Braunschweig, 1995. Google Scholar | DOI

[2] [2] Auslander, L., On radicals of discrete subgroups of Lie groups. Amer. J. Math. 85(1963), 145–150. Google Scholar | DOI

[3] [3] Auslander, L. and R. Tolimieri, On a conjecture of Mostow G. D. and the structure solvmanifolds. Bull. Amer. Math. Soc. 75(1969), 1330–1333. Google Scholar | DOI

[4] [4] Barth, W. and M. Otte, Uber fast-uniforme Untergruppen komplexer Liegruppen und auflosbare komplexe Mannigfaltigkeiten. Comment. Math. Helv. 44(1969), 269–281. Google Scholar | DOI

[5] [5] Barth, W., Invariante holomorphe Funktionen auf reduktiven Liegruppen. Math. Ann. 201(1973), 97–112. Google Scholar | DOI

[6] [6] Chevalley, C., Theorie des groupes deLie. Tome II. Groupes algebriques. Actualites Sci. Ind., 1152, Hermann & Cie., Paris, 1951. Google Scholar

[7] [7] Coeure, G. and J.-J. Loeb, A counterexample to the Serre problem with a bounded domain ofC2 as fiber. Ann. of Math. (2) 122(1985), no. 2, 329–334. Google Scholar | DOI

[8] [8] Gilligan, B., Ends of complex homogeneous manifolds having nonconstant holomorphic functions. Arch. Math. (Basel) 37(1981), no. 6, 544–555. Google Scholar | DOI

[9] [9] Gilligan, B. and Huckleberry, A. T., On non-compact complex nil-manifolds. Math. Ann. 238(1978), no. 1, 39–49. Google Scholar | DOI

[10] [10] Gilligan, B., C. Miebach, and K. Oeljeklaus, Pseudoconvex domains spread over complex homogeneous manifolds. Manuscripta Math. 142(2013),35-59. Google Scholar | DOI

[11] [11] Grauert, H., On Levi's problem and the imbedding of real-analytic manifolds. Ann. of Math. 68(1958), 460–472. Google Scholar | DOI

[12] [12] Hirschowitz, A., Le probleme de Levi pour les espaces homogenes. Bull. Soc. Math. France 103(1975), no. 2, 191–201. Google Scholar

[13] [13] Huckleberry, A. T. and E. Oeljeklaus, Homogeneous spaces from a complex analytic view-point. In: Manifolds and Lie Groups (Notre Dame, Ind., 1980) Progr. Math., 14, Birkhauser, Boston, MA, 1981, pp. 159–186. Google Scholar

[14] [14] Huckleberry, A. T., On holomorphically separable complex solv-manifolds. Ann. Inst. Fourier (Grenoble) 36(1986), no. 3, 57–65. Google Scholar | DOI

[15] [15] Jacobson, N., Lie algebras. Interscience Tracts in Pure and Applied Mathematics, 10, Interscience Publishers (a division of John Wiley & Sons), New York-London 1962. Google Scholar

[16] [16] Kiselman, C. O., The partial Legendre transformation for plurisubharmonic functions. Invent. Math. 49(1978), 137–148. Google Scholar | DOI

[17] [17] Matsushima, Y., On the discrete subgroups and homogeneous spaces ofnilpotent Lie groups. Nagoya Math. J. 2(1951), 95–110. Google Scholar | DOI

[18] [18] Matsushima, Y., Espaces homogenes de Stein des groupes de Lie complexes. Nagoya Math. J. 16(1960), 205–218. Google Scholar | DOI

[19] [19] Matsushima, Y. and A. Morimoto, Sur certains espaces fibres holomorphes sur une variete de Stein. Bull. Soc. Math. France 88(1960), 137–155. Google Scholar

[20] [20] Mostow, G. D., Factor spaces of solvable groups. Ann. of Math. (2) 60(1954), 1–27. Google Scholar | DOI

[21] [21] Mostow, G. D., Some applications of representative functions to solvmanifolds. Amer. J. Math. 93(1971), 11–32. Google Scholar | DOI

[22] [22] Narasimhan, R., The Levi problem for complex spaces. I, II, Math. Ann. 142(1961), 355–365; 146(1962), 195–216. Google Scholar | DOI

[23] [23] Narasimhan, R., The Levi problem in the theory of functions of several complex variables. Proc. Internat. Congr. Mathematicians (Stockholm, 1962), Inst. Mittag-Leffler, Djurholm, 1963, pp. 385–388. Google Scholar

[24] [24] Onishchik, A., Complex hulls of compact homogeneous spaces. Dokl. Akad. Nauk SSSR 130 (1960 ), 726– 729 (Russian); translation in Soviet Math. Dokl. 1(1960), 88–91. Google Scholar

[25] [25] Remmert, R., Sur les espaces analytiques holomorphiquement separables et holomorphiquement convexes. C. R. Acad. Sci. Paris 243(1956), 118–121. Google Scholar

[26] [26] Serre, J.-P., Quelques problemes globaux relatifs aux varietes de Stein. In: Colloque sur les fonctions de plusieurs variables, tenu a Bruxelles, 1953, Georges Thone, Liege; Masson & Cie, Paris, 1953, pp. 57–68. Google Scholar

[27] [27] Tits, J., Espaces homogenes complexes compacts. Comment. Math. Helv. 37(1962/1963), 111–120. Google Scholar | DOI