Given a Stein manifold which is homogeneous under a complex reductive Lie group , i.e., a complexification of a compact homogeneous space . Consider a relatively compact domain D which is invariant w.r.t. the compact real form G of the complex reductive Lie group in the Stein manifold . We find a relation between the automorphism group of the invariant domain D and isometric group of the compact homogeneous space . When the compact homogeneous space is isotropy irreducible, or even more general, we obtain a rigidity property of the automorphism groups.
Soit une variété de Stein qui est homogène sous un groupe de Lie réductif complexe , cést-à-dire, la complexification dʼun espace homogène compact . Soit D un domaine relativement compact qui est invariant par rapport à la forme compacte G de groupe de Lie réductif complexe dans . On trouve une relation entre le groupe dʼautomorphismes du domaine invariant D et le groupe dʼisométrie de lʼespace homogène compact . Si lʼespace homogène compact est isotropie irréductible, on obtient une propriété de rigidité du groupe dʼautomorphismes.
Accepted:
Published online:
Fusheng Deng 1; Xiangyu Zhou 2
@article{CRMATH_2012__350_7-8_417_0, author = {Fusheng Deng and Xiangyu Zhou}, title = {Rigidity of automorphism groups of invariant domains in certain {Stein} homogeneous manifolds}, journal = {Comptes Rendus. Math\'ematique}, pages = {417--420}, publisher = {Elsevier}, volume = {350}, number = {7-8}, year = {2012}, doi = {10.1016/j.crma.2012.02.009}, language = {en}, }
TY - JOUR AU - Fusheng Deng AU - Xiangyu Zhou TI - Rigidity of automorphism groups of invariant domains in certain Stein homogeneous manifolds JO - Comptes Rendus. Mathématique PY - 2012 SP - 417 EP - 420 VL - 350 IS - 7-8 PB - Elsevier DO - 10.1016/j.crma.2012.02.009 LA - en ID - CRMATH_2012__350_7-8_417_0 ER -
Fusheng Deng; Xiangyu Zhou. Rigidity of automorphism groups of invariant domains in certain Stein homogeneous manifolds. Comptes Rendus. Mathématique, Volume 350 (2012) no. 7-8, pp. 417-420. doi : 10.1016/j.crma.2012.02.009. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2012.02.009/
[1] Plurisubharmonic functions and Kählerian metrics on complexification of symmetric spaces, Indag. Math. N.S., Volume 3 (1992) no. 4, pp. 365-375
[2] Holomorphic equivalence and proper mapping of bounded Reinhardt domains not containing the origin, Comment. Math. Helv., Volume 59 (1984) no. 1, pp. 550-564
[3] On the automorphism group of a Stein manifold, Math. Ann., Volume 266 (1983), pp. 215-227
[4] Holomorphic mapping of products of annuli in , Pacific J. Math., Volume 87 (1980) no. 2, pp. 271-281
[5] Symplectic geometry and the uniqueness of Grauert tubes, Geom. Funct. Anal., Volume 11 (2001) no. 1, pp. 1-10
[6] On the uniqueness and characterization of Grauert tubes (V. Ancona; A. Silva, eds.), Complex Analysis and Geometry, 1995, pp. 119-133
[7] Univalence de certaines enveloppes dʼholomorphie, C. R. Acad. Sci. Paris, Sér. I, Volume 302 (1986) no. 2, pp. 59-61
[8] A counterexample to the Serre problem with a bounded domain of as fiber, Ann. of Math., Volume 122 (1985) no. 2, pp. 329-334
[9] F.S. Deng, X.Y. Zhou, Rigidity of automorphism groups of invariant domain in Stein homogeneous spaces, preprint.
[10] Invariant domains in complex symmetric spaces, J. Reine Angew. Math., Volume 454 (1994), pp. 97-118
[11] Hyperbolic manifolds whose envelopes of holomorphy are not hyperbolic | arXiv
[12] Lie transformation groups, Lie Groups and Lie Algebras, I, Springer, Berlin, 1993, pp. 95-235
[13] Grauert tubes and the homogeneous Monge–Ampère equation I, J. Diff. Geom., Volume 34 (1991), pp. 561-570
[14] Geometric invariant theory on Stein spaces, Math. Ann., Volume 281 (1991), pp. 631-662
[15] Hyperbolic Complex Spaces, Springer-Verlag, Berlin, Heidelberg, 1998
[16] Holomorphic automorphisms of hyperbolic Reinhardt domains, Math. USSR Izv., Volume 32 (1989) no. 1, pp. 15-37
[17] Séries de Laurent des fonctions holomorphes dans la complexifications dʼun espace symétrique compact, Ann. Sci. Ec. Norm. Sup., 4e série, Volume 11 (1978), pp. 167-210
[18] Global solutions of the homogeneous complex Monge–Ampère equation and complex structures on the tangent bundle of Riemannian manifolds, Math. Ann., Volume 290 (1991) no. 4, pp. 689-712
[19] Applications harmoniques et hyperbolicité de domaines tubes, Enseign. Math. (2), Volume 53 (2007) no. 3–4, pp. 347-367
[20] On complex automorphisms and holomorphic equivalence of domains, Symmetries in Complex Analysis, Amer. Math. Soc., Providence, RI, 2008, pp. 125-156
[21] Sur les automorphismes analytiques des variétés hyperboliques, Bull. Sci. Math., Volume 131 (2007) no. 5, pp. 469-476
[22] Homogeneous hyperbolic manifolds and homogeneous Siegel domains, J. Math. Kyoto Univ., Volume 25 (1985), pp. 269-291
[23] Stein manifolds with compact symmetric center, Math. Ann., Volume 289 (1991) no. 3, pp. 355-382
[24] Classical Topics in Complex Function Theory, Graduate Texts in Mathematics, vol. 172, Springer-Verlag, New York, 1998 (Translated from the German by Leslie Kay)
[25] Automorphisms and equivalence of bounded Reinhardt domains not containing the origin, Tohoku Math. J. (2), Volume 40 (1980) no. 1, pp. 119-152
[26] On classification and canonical realization of complex homogeneous bounded domains, Trudy Moskva Math. Obšč., Volume 12 (1963), pp. 359-388 (Translation in: Moscow Math. Soc., 12, 1963, pp. 404-437)
[27] On isotropy irreducible Riemannian manifolds, Acta Math., Volume 166 (1991), pp. 223-261
[28] The geometry and structure of isotropy irreducible homogeneous spaces, Acta Math., Volume 120 (1968), pp. 59-148 (correction Acta Math., 152, 1984, pp. 141-142)
[29] On orbit connectedness, orbit convexity, and envelope of holomorphy, Izv. Russian Akad. Nauk, Ser. Math., Volume 58 (1994) no. 2, pp. 196-205
[30] On invariant domains in certain complex homogeneous spaces, Ann. Inst. Fourier, Volume 47 (1997) no. 4, pp. 1101-1115
[31] Some results related to group actions in several complex variables, Beijing, 2002 (Proceedings of the International Congress of Mathematicians), Volume vol. II, Higher Education Press, Beijing (2002), pp. 743-753
Cited by Sources:
Comments - Policy