## Invariant domains of holomorphy: twenty years later.(English. Russian original)Zbl 1302.32011

Proc. Steklov Inst. Math. 285, 241-250 (2014); translation from Tr. Mat. Inst. Steklova 285, 253-263 (2014).
Summary: 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 [Proc. Steklov Inst. Math. 203, 145–155 (1995); translation from Tr. Mat. Inst. Steklova 203, 159–172 (1994; Zbl 0911.32020)] 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.

### MSC:

 32D05 Domains of holomorphy 32M05 Complex Lie groups, group actions on complex spaces

Zbl 0911.32020
Full Text:

### References:

 [1] Barrett, D E, Holomorphic equivalence and proper mapping of bounded Reinhardt domains not containing the origin, Comment. Math. Helv., 59, 550-564, (1984) · Zbl 0601.32030 [2] Bedford, E, Holomorphic mapping of products of annuli in C\^{}{n}, Pac. J. Math., 87, 271-281, (1980) · Zbl 0449.32024 [3] Deng, F; Zhou, X, Rigidity of automorphism groups of invariant domains in certain Stein homogeneous manifolds, C. R., Math., Acad. Sci. Paris, 350, 417-420, (2012) · Zbl 1244.32003 [4] Deng, F; Zhou, X, Rigidity of automorphism groups of invariant domains in homogeneous Stein spaces, Izv. Ross. Akad. Nauk, Ser. Mat., 78, 37-64, (2014) · Zbl 1293.14010 [5] Fels, G; Geatti, L, Invariant domains in complex symmetric spaces, J. Reine Angew. Math., 454, 97-118, (1994) · Zbl 0803.32019 [6] Heinzner, P, Geometric invariant theory on Stein spaces, Math. Ann., 289, 631-662, (1991) · Zbl 0728.32010 [7] Kiselman, C O, The partial Legendre transformation for plurisubharmonic functions, Invent. Math., 49, 137-148, (1978) · Zbl 0378.32010 [8] Kruzhilin, N G, Holomorphic automorphisms of hyperbolic Reinhardt domains, Izv. Akad. Nauk SSSR, Ser. Mat., 52, 16-40, (1988) · Zbl 0638.32024 [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). · Zbl 0563.32013 [10] Loeb, J-J, Pseudo-convexité des ouverts invariants et convexité geodésique dans certains espaces symétriques, No. 1198, 172-190, (1986), Berlin · Zbl 0595.32024 [11] D. Luna, “Slices étales,” Bull. Soc. Math. France 33, 81-105 (1973). · Zbl 0286.14014 [12] Sergeev, A G; Heinzner, P, The extended matrix disc is a domain of holomorphy, Izv. Akad. Nauk SSSR, Ser. Mat., 55, 647-657, (1991) · Zbl 0754.32006 [13] Sergeev, A G; Zhou, X, On invariant domains of holomorphy, Tr. Mat. Inst. im. V.A. Steklova, Ross. Akad. Nauk, 203, 159-172, (1994) · Zbl 0911.32020 [14] Sergeev, A G; Zhou, X, Extended future tube conjecture, Tr. Mat. Inst. im. V.A. Steklova, Ross. Akad. Nauk, 228, 32-51, (2000) · Zbl 0994.32008 [15] Shimizu, S, Automorphisms and equivalence of bounded Reinhardt domains not containing the origin, Tohoku Math. J., Ser. 2, 40, 119-152, (1988) · Zbl 0646.32003 [16] Snow, D M, Reductive group actions on Stein spaces, Math. Ann., 259, 79-97, (1982) · Zbl 0509.32021 [17] Szöke, R, Complex structures on tangent bundles of Riemannian manifolds, Math. Ann., 291, 409-428, (1991) · Zbl 0749.53021 [18] Szöke, R, Automorphisms of certain Stein manifolds, Math. Z., 219, 357-385, (1995) · Zbl 0829.32009 [19] Vladimirov, V S, Nikolai nikolaevich bogolyubov-Mathematician by the grace of god, 475-499, (2006), Berlin · Zbl 1086.01034 [20] J. A. Wolf, “The geometry and structure of isotropy irreducible homogeneous spaces,” Acta Math. 120, 59-148 · Zbl 0157.52102 [21] Zhou, X, On orbit connectedness, orbit convexity, and envelopes of holomorphy, Izv. Ross. Akad. Nauk, Ser. Mat., 58, 196-205, (1994) · Zbl 0835.32006 [22] Zhou, X, On invariant domains in certain complex homogeneous spaces, Ann. Inst. Fourier, 47, 1101-1115, (1997) · Zbl 0881.32015 [23] Zhou, X, The extended future tube is a domain of holomorphy, Math. Res. Lett., 5, 185-190, (1998) · Zbl 0914.32002 [24] Zhou, X, A proof of the extended future tube conjecture, Izv. Ross. Akad. Nauk, Ser. Mat., 62, 211-224, (1998) [25] Zhou, X, An invariant version of cartan’s lemma and complexification of invariant domains of holomorphy, Dokl. Akad. Nauk, 366, 608-612, (1999) · Zbl 0967.32021
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.