Advertisement

Invariant domains of holomorphy: Twenty years later

  • A. G. Sergeev
  • Xiangyu Zhou
Article
  • 74 Downloads

Abstract

This review is devoted to the domains of holomorphy invariant under holomorphic actions of real Lie groups. We have collected here the results on this subject obtained during the last twenty years, which have passed since the publication of the first review of the authors on this topic. This first review was mainly devoted to the case of compact transformation groups, while the first two sections of the present review deal mostly with noncompact groups. In Section 3 we discuss the problem of rigidity of automorphism groups of domains of holomorphy invariant under compact transformation groups.

Keywords

STEKLOV Institute Stein Manifold Holomorphic Extension Invariant Domain Reinhardt Domain 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    D. E. Barrett, “Holomorphic equivalence and proper mapping of bounded Reinhardt domains not containing the origin,” Comment. Math. Helv. 59, 550–564 (1984).CrossRefMathSciNetGoogle Scholar
  2. 2.
    E. Bedford, “Holomorphic mapping of products of annuli in C n,” Pac. J. Math. 87(2), 271–281 (1980).CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    F. Deng and X. Zhou, “Rigidity of automorphism groups of invariant domains in certain Stein homogeneous manifolds,” C. R., Math., Acad. Sci. Paris 350, 417–420 (2012).CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    F. Deng and X. Zhou, “Rigidity of automorphism groups of invariant domains in homogeneous Stein spaces,” Izv. Ross. Akad. Nauk, Ser. Mat. 78(1), 37–64 (2014) [Izv. Math. 78, 34–58 (2014)].CrossRefMathSciNetGoogle Scholar
  5. 5.
    G. Fels and L. Geatti, “Invariant domains in complex symmetric spaces,” J. Reine Angew. Math. 454, 97–118 (1994).MATHMathSciNetGoogle Scholar
  6. 6.
    P. Heinzner, “Geometric invariant theory on Stein spaces,” Math. Ann. 289, 631–662 (1991).CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    C. O. Kiselman, “The partial Legendre transformation for plurisubharmonic functions,” Invent. Math. 49, 137–148 (1978).CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    N. G. Kruzhilin, “Holomorphic automorphisms of hyperbolic Reinhardt domains,” Izv. Akad. Nauk SSSR, Ser. Mat. 52(1), 16–40 (1988) [Math. USSR, Izv. 32, 15–38 (1989)].MATHMathSciNetGoogle Scholar
  9. 9.
    J.-J. Loeb, “Action d’une forme réelle d’un groupe de Lie complexe sur les fonctions plurisousharmoniques,” Ann. Inst. Fourier 35(4), 59–97 (1985).CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    J.-J. Loeb, “Pseudo-convexité des ouverts invariants et convexité geodésique dans certains espaces symétriques,” in Séminaire d’analyse P. Lelong-P. Dolbeault-H. Skoda, Années 1983/1984 (Springer, Berlin, 1986), Lect. Notes Math. 1198, pp. 172–190.Google Scholar
  11. 11.
    D. Luna, “Slices étales,” Bull. Soc. Math. France 33, 81–105 (1973).MATHGoogle Scholar
  12. 12.
    A. G. Sergeev and P. Heinzner, “The extended matrix disc is a domain of holomorphy,” Izv. Akad. Nauk SSSR, Ser. Mat. 55(3), 647–657 (1991) [Math. USSR, Izv. 38, 637–645 (1992)].MATHMathSciNetGoogle Scholar
  13. 13.
    A. G. Sergeev and X. Zhou, “On invariant domains of holomorphy,” Tr. Mat. Inst. im. V.A. Steklova, Ross. Akad. Nauk 203, 159–172 (1994) [Proc. Steklov Inst. Math. 203, 145–155 (1995)].Google Scholar
  14. 14.
    A. G. Sergeev and X. Zhou, “Extended future tube conjecture,” Tr. Mat. Inst. im. V.A. Steklova, Ross. Akad. Nauk 228, 32–51 (2000) [Proc. Steklov Inst. Math. 228, 25–42 (2000)].MathSciNetGoogle Scholar
  15. 15.
    S. Shimizu, “Automorphisms and equivalence of bounded Reinhardt domains not containing the origin,” Tohoku Math. J., Ser. 2, 40(1), 119–152 (1988).CrossRefMATHGoogle Scholar
  16. 16.
    D. M. Snow, “Reductive group actions on Stein spaces,” Math. Ann. 259, 79–97 (1982).CrossRefMATHMathSciNetGoogle Scholar
  17. 17.
    R. Szöke,, “Complex structures on tangent bundles of Riemannian manifolds,” Math. Ann. 291, 409–428 (1991).CrossRefMATHMathSciNetGoogle Scholar
  18. 18.
    R. Szöke,, “Automorphisms of certain Stein manifolds,” Math. Z. 219(3), 357–385 (1995).CrossRefMATHMathSciNetGoogle Scholar
  19. 19.
    V. S. Vladimirov, “Nikolai Nikolaevich Bogolyubov-mathematician by the grace of God,” in Mathematical Events of the Twentieth Century (Springer, Berlin, 2006), pp. 475–499.CrossRefGoogle Scholar
  20. 20.
    J. A. Wolf, “The geometry and structure of isotropy irreducible homogeneous spaces,” Acta Math. 120, 59–148Google Scholar
  21. 21.
    X. Zhou, “On orbit connectedness, orbit convexity, and envelopes of holomorphy,” Izv. Ross. Akad. Nauk, Ser. Mat. 58(2), 196–205 (1994) [Russ. Acad. Sci., Izv. Math. 44, 403–413 (1995)].Google Scholar
  22. 22.
    X. Zhou, “On invariant domains in certain complex homogeneous spaces,” Ann. Inst. Fourier 47(4), 1101–1115 (1997).CrossRefMATHMathSciNetGoogle Scholar
  23. 23.
    X. Zhou, “The extended future tube is a domain of holomorphy,” Math. Res. Lett. 5, 185–190 (1998).CrossRefMATHMathSciNetGoogle Scholar
  24. 24.
    X. Zhou, “A proof of the extended future tube conjecture,” Izv. Ross. Akad. Nauk, Ser. Mat. 62(1), 211–224 (1998) [Izv. Math. 62, 201–213 (1998)].CrossRefMathSciNetGoogle Scholar
  25. 25.
    X. Zhou, “An invariant version of Cartan’s lemma and complexification of invariant domains of holomorphy,” Dokl. Akad. Nauk 366(5), 608–612 (1999) [Dokl. Math. 59 (3), 460–463 (1999)].MathSciNetGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2014

Authors and Affiliations

  1. 1.Steklov Mathematical InstituteRussian Academy of SciencesMoscowRussia
  2. 2.Institute of MathematicsAcademy of Mathematics and Systems ScienceBeijingChina
  3. 3.Hua Loo-Keng Key Laboratory of MathematicsChinese Academy of SciencesBeijingChina

Personalised recommendations