Skip to main content
SpringerLink
Log in

Invariant domains of holomorphy: Twenty years later

  • Published:
Proceedings of the Steklov Institute of Mathematics Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. D. E. Barrett, “Holomorphic equivalence and proper mapping of bounded Reinhardt domains not containing the origin,” Comment. Math. Helv. 59, 550–564 (1984).

    Article  MathSciNet  Google Scholar 

  2. E. Bedford, “Holomorphic mapping of products of annuli in C n,” Pac. J. Math. 87(2), 271–281 (1980).

    Article  MATH  MathSciNet  Google Scholar 

  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).

    Article  MATH  MathSciNet  Google Scholar 

  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)].

    Article  MathSciNet  Google Scholar 

  5. G. Fels and L. Geatti, “Invariant domains in complex symmetric spaces,” J. Reine Angew. Math. 454, 97–118 (1994).

    MATH  MathSciNet  Google Scholar 

  6. P. Heinzner, “Geometric invariant theory on Stein spaces,” Math. Ann. 289, 631–662 (1991).

    Article  MATH  MathSciNet  Google Scholar 

  7. C. O. Kiselman, “The partial Legendre transformation for plurisubharmonic functions,” Invent. Math. 49, 137–148 (1978).

    Article  MATH  MathSciNet  Google Scholar 

  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)].

    MATH  MathSciNet  Google Scholar 

  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).

    Article  MATH  MathSciNet  Google Scholar 

  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. D. Luna, “Slices étales,” Bull. Soc. Math. France 33, 81–105 (1973).

    MATH  Google Scholar 

  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)].

    MATH  MathSciNet  Google Scholar 

  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. 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)].

    MathSciNet  Google Scholar 

  15. S. Shimizu, “Automorphisms and equivalence of bounded Reinhardt domains not containing the origin,” Tohoku Math. J., Ser. 2, 40(1), 119–152 (1988).

    Article  MATH  Google Scholar 

  16. D. M. Snow, “Reductive group actions on Stein spaces,” Math. Ann. 259, 79–97 (1982).

    Article  MATH  MathSciNet  Google Scholar 

  17. R. Szöke,, “Complex structures on tangent bundles of Riemannian manifolds,” Math. Ann. 291, 409–428 (1991).

    Article  MATH  MathSciNet  Google Scholar 

  18. R. Szöke,, “Automorphisms of certain Stein manifolds,” Math. Z. 219(3), 357–385 (1995).

    Article  MATH  MathSciNet  Google Scholar 

  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.

    Chapter  Google Scholar 

  20. J. A. Wolf, “The geometry and structure of isotropy irreducible homogeneous spaces,” Acta Math. 120, 59–148

  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. X. Zhou, “On invariant domains in certain complex homogeneous spaces,” Ann. Inst. Fourier 47(4), 1101–1115 (1997).

    Article  MATH  MathSciNet  Google Scholar 

  23. X. Zhou, “The extended future tube is a domain of holomorphy,” Math. Res. Lett. 5, 185–190 (1998).

    Article  MATH  MathSciNet  Google Scholar 

  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)].

    Article  MathSciNet  Google Scholar 

  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)].

    MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Steklov Mathematical Institute, Russian Academy of Sciences, ul. Gubkina 8, Moscow, 119991, Russia

    A. G. Sergeev

  2. Institute of Mathematics, Academy of Mathematics and Systems Science, Beijing, China

    Xiangyu Zhou

  3. Hua Loo-Keng Key Laboratory of Mathematics, Chinese Academy of Sciences, Beijing, China

    Xiangyu Zhou

Authors
  1. A. G. Sergeev
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Xiangyu Zhou
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. G. Sergeev.

Additional information

Published in Russian in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2014, Vol. 285, pp. 253–263.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Sergeev, A.G., Zhou, X. Invariant domains of holomorphy: Twenty years later. Proc. Steklov Inst. Math. 285, 241–250 (2014). https://doi.org/10.1134/S0081543814040178

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0081543814040178

Keywords

Search

Navigation