Invariant meromorphic functions on Stein spaces

Daniel Greb[1]; Christian Miebach[2]

  • [1] Albert-Ludwigs-Universität Freiburg Mathematisches Institut Abteilung für Reine Mathematik Eckerstr. 1 79104 Freiburg im Breisgau Germany
  • [2] Laboratoire de Mathématiques Pures et Appliquées Université du Littoral 50, rue F. Buisson 62228 Calais Cedex France

Annales de l’institut Fourier (2012)

  • Volume: 62, Issue: 5, page 1983-2011
  • ISSN: 0373-0956

Abstract

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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.

How to cite

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Greb, Daniel, and Miebach, Christian. "Invariant meromorphic functions on Stein spaces." Annales de l’institut Fourier 62.5 (2012): 1983-2011. <http://eudml.org/doc/251086>.

@article{Greb2012,
abstract = {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.},
affiliation = {Albert-Ludwigs-Universität Freiburg Mathematisches Institut Abteilung für Reine Mathematik Eckerstr. 1 79104 Freiburg im Breisgau Germany; Laboratoire de Mathématiques Pures et Appliquées Université du Littoral 50, rue F. Buisson 62228 Calais Cedex France},
author = {Greb, Daniel, Miebach, Christian},
journal = {Annales de l’institut Fourier},
keywords = {Lie group action; Stein space; invariant meromorphic function; Rosenlicht quotient},
language = {eng},
number = {5},
pages = {1983-2011},
publisher = {Association des Annales de l’institut Fourier},
title = {Invariant meromorphic functions on Stein spaces},
url = {http://eudml.org/doc/251086},
volume = {62},
year = {2012},
}

TY - JOUR
AU - Greb, Daniel
AU - Miebach, Christian
TI - Invariant meromorphic functions on Stein spaces
JO - Annales de l’institut Fourier
PY - 2012
PB - Association des Annales de l’institut Fourier
VL - 62
IS - 5
SP - 1983
EP - 2011
AB - 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.
LA - eng
KW - Lie group action; Stein space; invariant meromorphic function; Rosenlicht quotient
UR - http://eudml.org/doc/251086
ER -

References

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  1. D. N. Akhiezer, Invariant meromorphic functions on complex semisimple Lie groups, Invent. Math. 65 (1981/82), 325-329 Zbl0479.32010MR643557
  2. Andrzej Białynicki-Birula, Quotients by actions of groups, Algebraic quotients. Torus actions and cohomology. The adjoint representation and the adjoint action 131 (2002), 1-82, Springer, Berlin Zbl1061.14046MR1925828
  3. David Birkes, Orbits of linear algebraic groups, Ann. of Math. (2) 93 (1971), 459-475 Zbl0198.35001MR296077
  4. Gerd Fischer, Complex analytic geometry, (1976), Springer-Verlag, Berlin Zbl0343.32002MR430286
  5. Akira Fujiki, On automorphism groups of compact Kähler manifolds, Invent. Math. 44 (1978), 225-258 Zbl0367.32004MR481142
  6. Hans Grauert, Reinhold Remmert, Theory of Stein spaces, 236 (1979), Springer-Verlag, Berlin Zbl0433.32007MR580152
  7. Hans Grauert, Reinhold Remmert, Coherent analytic sheaves, 265 (1984), Springer-Verlag, Berlin Zbl0537.32001MR755331
  8. Daniel Greb, Compact Kähler quotients of algebraic varieties and Geometric Invariant Theory, Adv. Math. 224 (2010), 401-431 Zbl1216.14044MR2609010
  9. Daniel Greb, Projectivity of analytic Hilbert and Kähler quotients, Trans. Amer. Math. Soc. 362 (2010), 3243-3271 Zbl1216.14045MR2592955
  10. Peter Heinzner, Linear äquivariante Einbettungen Steinscher Räume, Math. Ann. 280 (1988), 147-160 Zbl0617.32022MR928302
  11. Peter Heinzner, Geometric invariant theory on Stein spaces, Math. Ann. 289 (1991), 631-662 Zbl0728.32010MR1103041
  12. Peter Heinzner, Luca Migliorini, Marzia Polito, Semistable quotients, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 26 (1998), 233-248 Zbl0922.32017MR1631577
  13. Harald Holmann, Komplexe Räume mit komplexen Transformations-gruppen, Math. Ann. 150 (1963), 327-360 Zbl0156.30603MR150789
  14. John H. Hubbard, Ralph W. Oberste-Vorth, Hénon mappings in the complex domain. I. The global topology of dynamical space, Inst. Hautes Études Sci. Publ. Math. (1994), 5-46 Zbl0839.54029MR1307296
  15. A. Huckleberry, E. Oeljeklaus, Classification theorems for almost homogeneous spaces, 9 (1984), Université de Nancy Institut Élie Cartan, Nancy Zbl0549.32024MR782881
  16. David I. Lieberman, Compactness of the Chow scheme: applications to automorphisms and deformations of Kähler manifolds, Fonctions de plusieurs variables complexes, III (Sém. François Norguet, 1975–1977) 670 (1978), 140-186, Springer, Berlin Zbl0391.32018MR521918
  17. Domingo Luna, Slices étales, Sur les groupes algébriques (1973), 81-105. Bull. Soc. Math. France, Paris, Mémoire 33, Soc. Math. France, Paris Zbl0286.14014MR318167
  18. Domingo Luna, Fonctions différentiables invariantes sous l’opération d’un groupe réductif, Ann. Inst. Fourier (Grenoble) 26 (1976), ix, 33-49 Zbl0315.20039MR423398
  19. Raghavan Narasimhan, Imbedding of holomorphically complete complex spaces, Amer. J. Math. 82 (1960), 917-934 Zbl0104.05402MR148942
  20. V. L. Popov, È. B. Vinberg, Invariant theory, Algebraic geometry IV 55 (1994), 123-284, Springer-Verlag, Berlin Zbl0789.14008
  21. Zinovy Reichstein, Nikolaus Vonessen, Stable affine models for algebraic group actions, J. Lie Theory 14 (2004), 563-568 Zbl1060.14067MR2066872
  22. Reinhold Remmert, Holomorphe und meromorphe Abbildungen komplexer Räume, Math. Ann. 133 (1957), 328-370 Zbl0079.10201MR92996
  23. R. W. Richardson, Deformations of Lie subgroups and the variation of isotropy subgroups, Acta Math. 129 (1972), 35-73 Zbl0242.22020MR299723
  24. R. W. Richardson, Principle orbit types for reductive groups acting on Stein manifolds, Math. Ann. 208 (1974), 323-331 Zbl0267.32015MR355123
  25. Maxwell Rosenlicht, Some basic theorems on algebraic groups, Amer. J. Math. 78 (1956), 401-443 Zbl0073.37601MR82183
  26. Dennis M. Snow, Reductive group actions on Stein spaces, Math. Ann. 259 (1982), 79-97 Zbl0509.32021MR656653
  27. Wilhelm Stoll, Über meromorphe Abbildungen komplexer Räume. I, Math. Ann. 136 (1958), 201-239 Zbl0096.06202MR103283
  28. Wilhelm Stoll, Über meromorphe Abbildungen komplexer Räume. II, Math. Ann. 136 (1958), 393-429 Zbl0096.06202MR103284
  29. Jean-Louis Verdier, Stratifications de Whitney et théorème de Bertini-Sard, Invent. Math. 36 (1976), 295-312 Zbl0333.32010MR481096

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