[Fonctions méromorphes invariantes sur les espaces de Stein]
Dans ce travail nous développons des outils et des méthodes fondamentaux afin d’étudier les fonctions méromorphes invariantes sur les espaces de Stein munis d’une action holomorphe d’un groupe complexe-réductif . Nous construisons des quotients à la Rosenlicht pour l’action d’un sous-groupe algébrique de sur . En particulier on montre que dans cette situation les fonctions méromorphes invariantes sous ce sous-groupe algébrique séparent ses orbites en position générale. Nous donnons aussi des applications concernant les espaces presque homogènes et les types d’orbite principaux. De plus, le résultat principal est utilisé afin de clarifier la relation entre les invariants holomorphes voire méromorphes de . Une étape importante de notre preuve consiste à montrer un analogue faible équivariant du théorème de Narasimhan sur les plongements propres des espaces de Stein.
In this paper we develop fundamental tools and methods to study meromorphic functions in an equivariant setup. As our main result we construct quotients of Rosenlicht-type for Stein spaces acted upon holomorphically by complex-reductive Lie groups and their algebraic subgroups. In particular, we show that in this setup invariant meromorphic functions separate orbits in general position. Applications to almost homogeneous spaces and principal orbit types are given. Furthermore, we use the main result to investigate the relation between holomorphic and meromorphic invariants for reductive group actions. As one important step in our proof we obtain a weak equivariant analogue of Narasimhan’s embedding theorem for Stein spaces.
Keywords: Lie group action, Stein space, invariant meromorphic function, Rosenlicht quotient
Mot clés : action des groupes de Lie, espace de Stein, fonctions méromorphes invariantes, quotient à la Rosenlicht
@article{AIF_2012__62_5_1983_0,
author = {Greb, Daniel and Miebach, Christian},
title = {Invariant meromorphic functions on {Stein} spaces},
journal = {Annales de l'Institut Fourier},
pages = {1983--2011},
publisher = {Association des Annales de l{\textquoteright}institut Fourier},
volume = {62},
number = {5},
year = {2012},
doi = {10.5802/aif.2740},
zbl = {1270.32005},
mrnumber = {3025158},
language = {en},
url = {http://www.numdam.org/articles/10.5802/aif.2740/}
}
TY - JOUR AU - Greb, Daniel AU - Miebach, Christian TI - Invariant meromorphic functions on Stein spaces JO - Annales de l'Institut Fourier PY - 2012 SP - 1983 EP - 2011 VL - 62 IS - 5 PB - Association des Annales de l’institut Fourier UR - http://www.numdam.org/articles/10.5802/aif.2740/ DO - 10.5802/aif.2740 LA - en ID - AIF_2012__62_5_1983_0 ER -
%0 Journal Article %A Greb, Daniel %A Miebach, Christian %T Invariant meromorphic functions on Stein spaces %J Annales de l'Institut Fourier %D 2012 %P 1983-2011 %V 62 %N 5 %I Association des Annales de l’institut Fourier %U http://www.numdam.org/articles/10.5802/aif.2740/ %R 10.5802/aif.2740 %G en %F AIF_2012__62_5_1983_0
Greb, Daniel; Miebach, Christian. Invariant meromorphic functions on Stein spaces. Annales de l'Institut Fourier, Tome 62 (2012) no. 5, pp. 1983-2011. doi : 10.5802/aif.2740. http://www.numdam.org/articles/10.5802/aif.2740/
[1] Invariant meromorphic functions on complex semisimple Lie groups, Invent. Math., Volume 65 (1981/82) no. 3, pp. 325-329 | DOI | MR | Zbl
[2] Quotients by actions of groups, Algebraic quotients. Torus actions and cohomology. The adjoint representation and the adjoint action (Encyclopaedia Math. Sci.), Volume 131, Springer, Berlin, 2002, pp. 1-82 | MR | Zbl
[3] Orbits of linear algebraic groups, Ann. of Math. (2), Volume 93 (1971), pp. 459-475 | DOI | MR | Zbl
[4] Complex analytic geometry, Lecture Notes in Mathematics, Vol. 538, Springer-Verlag, Berlin, 1976 | MR | Zbl
[5] On automorphism groups of compact Kähler manifolds, Invent. Math., Volume 44 (1978) no. 3, pp. 225-258 | DOI | MR | Zbl
[6] Theory of Stein spaces, Grundlehren der Mathematischen Wissenschaften, 236, Springer-Verlag, Berlin, 1979 (Translated from the German by Alan Huckleberry) | MR | Zbl
[7] Coherent analytic sheaves, Grundlehren der Mathematischen Wissenschaften, 265, Springer-Verlag, Berlin, 1984 | MR | Zbl
[8] Compact Kähler quotients of algebraic varieties and Geometric Invariant Theory, Adv. Math., Volume 224 (2010) no. 2, pp. 401-431 | DOI | MR | Zbl
[9] Projectivity of analytic Hilbert and Kähler quotients, Trans. Amer. Math. Soc., Volume 362 (2010) no. 6, pp. 3243-3271 | DOI | MR | Zbl
[10] Linear äquivariante Einbettungen Steinscher Räume, Math. Ann., Volume 280 (1988) no. 1, pp. 147-160 | DOI | MR | Zbl
[11] Geometric invariant theory on Stein spaces, Math. Ann., Volume 289 (1991) no. 4, pp. 631-662 | DOI | MR | Zbl
[12] Semistable quotients, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), Volume 26 (1998) no. 2, pp. 233-248 | Numdam | MR | Zbl
[13] Komplexe Räume mit komplexen Transformations-gruppen, Math. Ann., Volume 150 (1963), pp. 327-360 | DOI | MR | Zbl
[14] Hénon mappings in the complex domain. I. The global topology of dynamical space, Inst. Hautes Études Sci. Publ. Math. (1994) no. 79, pp. 5-46 | Numdam | MR | Zbl
[15] Classification theorems for almost homogeneous spaces, Institut Élie Cartan, 9, Université de Nancy Institut Élie Cartan, Nancy, 1984 | MR | Zbl
[16] Compactness of the Chow scheme: applications to automorphisms and deformations of Kähler manifolds, Fonctions de plusieurs variables complexes, III (Sém. François Norguet, 1975–1977) (Lecture Notes in Math.), Volume 670, Springer, Berlin, 1978, pp. 140-186 | MR | Zbl
[17] Slices étales, Sur les groupes algébriques, Soc. Math. France, Paris, 1973, p. 81-105. Bull. Soc. Math. France, Paris, Mémoire 33 | Numdam | MR | Zbl
[18] Fonctions différentiables invariantes sous l’opération d’un groupe réductif, Ann. Inst. Fourier (Grenoble), Volume 26 (1976) no. 1, pp. ix, 33-49 | DOI | Numdam | MR | Zbl
[19] Imbedding of holomorphically complete complex spaces, Amer. J. Math., Volume 82 (1960), pp. 917-934 | DOI | MR | Zbl
[20] Invariant theory, Algebraic geometry IV (Encyclopaedia of Mathematical Sciences), Volume 55, Springer-Verlag, Berlin, 1994, pp. 123-284 | Zbl
[21] Stable affine models for algebraic group actions, J. Lie Theory, Volume 14 (2004) no. 2, pp. 563-568 | MR | Zbl
[22] Holomorphe und meromorphe Abbildungen komplexer Räume, Math. Ann., Volume 133 (1957), pp. 328-370 | DOI | MR | Zbl
[23] Deformations of Lie subgroups and the variation of isotropy subgroups, Acta Math., Volume 129 (1972), pp. 35-73 | DOI | MR | Zbl
[24] Principle orbit types for reductive groups acting on Stein manifolds, Math. Ann., Volume 208 (1974), pp. 323-331 | DOI | MR | Zbl
[25] Some basic theorems on algebraic groups, Amer. J. Math., Volume 78 (1956), pp. 401-443 | DOI | MR | Zbl
[26] Reductive group actions on Stein spaces, Math. Ann., Volume 259 (1982) no. 1, pp. 79-97 | DOI | MR | Zbl
[27] Über meromorphe Abbildungen komplexer Räume. I, Math. Ann., Volume 136 (1958), pp. 201-239 | DOI | MR | Zbl
[28] Über meromorphe Abbildungen komplexer Räume. II, Math. Ann., Volume 136 (1958), pp. 393-429 | DOI | MR | Zbl
[29] Stratifications de Whitney et théorème de Bertini-Sard, Invent. Math., Volume 36 (1976), pp. 295-312 | DOI | MR | Zbl
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