Rigidity of automorphism groups of invariant domains in certain Stein homogeneous manifolds

Deng, Fusheng; Zhou, Xiangyu

Comptes Rendus. Mathématique, Tome 350 (2012) no. 7-8, pp. 417-420.

Soit XC une variété de Stein qui est homogène sous un groupe de Lie réductif complexe GC, cést-à-dire, la complexification GC/KC dʼun espace homogène compact G/K. Soit D un domaine relativement compact qui est invariant par rapport à la forme compacte G de groupe de Lie réductif complexe dans XC. On trouve une relation entre le groupe dʼautomorphismes du domaine invariant D et le groupe dʼisométrie de lʼespace homogène compact G/K. Si lʼespace homogène compact G/K est isotropie irréductible, on obtient une propriété de rigidité du groupe dʼautomorphismes.

Given a Stein manifold XC which is homogeneous under a complex reductive Lie group GC, i.e., a complexification GC/KC of a compact homogeneous space G/K. 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 XC. We find a relation between the automorphism group of the invariant domain D and isometric group of the compact homogeneous space G/K. When the compact homogeneous space G/K is isotropy irreducible, or even more general, we obtain a rigidity property of the automorphism groups.

[1] Azad, H.; Loeb, J.-J. Plurisubharmonic functions and Kählerian metrics on complexification of symmetric spaces, Indag. Math. N.S., Volume 3 (1992) no. 4, pp. 365-375

[2] Barrett, D.E. 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] Bedford, E. On the automorphism group of a Stein manifold, Math. Ann., Volume 266 (1983), pp. 215-227

[4] Bedford, E. Holomorphic mapping of products of annuli in Cn, Pacific J. Math., Volume 87 (1980) no. 2, pp. 271-281

[5] Burns, D.; Hind, R. Symplectic geometry and the uniqueness of Grauert tubes, Geom. Funct. Anal., Volume 11 (2001) no. 1, pp. 1-10

[6] Burns, D. On the uniqueness and characterization of Grauert tubes (Ancona, V.; Silva, A., eds.), Complex Analysis and Geometry, 1995, pp. 119-133

[7] Coeuré, G.; Loeb, J.-J. Univalence de certaines enveloppes dʼholomorphie, C. R. Acad. Sci. Paris, Sér. I, Volume 302 (1986) no. 2, pp. 59-61

[8] Coeuré, G.; Loeb, J.-J. A counterexample to the Serre problem with a bounded domain of C2 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] Fels, G.; Geatti, L. Invariant domains in complex symmetric spaces, J. Reine Angew. Math., Volume 454 (1994), pp. 97-118

[11] Geatti, L.; Iannuzzi, A.; Loeb, J.-J. Hyperbolic manifolds whose envelopes of holomorphy are not hyperbolic | arXiv

[12] Gorbatsevich, V.V.; Onishchik, A.L. Lie transformation groups, Lie Groups and Lie Algebras, I, Springer, Berlin, 1993, pp. 95-235

[13] Guillemin, V.; Stenzel, M. Grauert tubes and the homogeneous Monge–Ampère equation I, J. Diff. Geom., Volume 34 (1991), pp. 561-570

[14] Heinzner, P. Geometric invariant theory on Stein spaces, Math. Ann., Volume 281 (1991), pp. 631-662

[15] Kobayashi, S. Hyperbolic Complex Spaces, Springer-Verlag, Berlin, Heidelberg, 1998

[16] Kruzhilin, N.G. Holomorphic automorphisms of hyperbolic Reinhardt domains, Math. USSR Izv., Volume 32 (1989) no. 1, pp. 15-37

[17] Lassalle, M. 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] Lempert, L.; Szöke, R. 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] Loeb, J.J. Applications harmoniques et hyperbolicité de domaines tubes, Enseign. Math. (2), Volume 53 (2007) no. 3–4, pp. 347-367

[20] Loeb, J.-J. On complex automorphisms and holomorphic equivalence of domains, Symmetries in Complex Analysis, Amer. Math. Soc., Providence, RI, 2008, pp. 125-156

[21] Loeb, J.-J.; Vigué, J.-P. Sur les automorphismes analytiques des variétés hyperboliques, Bull. Sci. Math., Volume 131 (2007) no. 5, pp. 469-476

[22] Nakajima, K. Homogeneous hyperbolic manifolds and homogeneous Siegel domains, J. Math. Kyoto Univ., Volume 25 (1985), pp. 269-291

[23] Patrizio, Giorgio; Wong, Pit-Mann Stein manifolds with compact symmetric center, Math. Ann., Volume 289 (1991) no. 3, pp. 355-382

[24] Remmert, R. 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] Shimizu, S. Automorphisms and equivalence of bounded Reinhardt domains not containing the origin, Tohoku Math. J. (2), Volume 40 (1980) no. 1, pp. 119-152

[26] Vinberg, E.B.; Gindikin, S.G.; Piatetski-Sapiro, I.I. 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] Wang, M.; Ziller, W. On isotropy irreducible Riemannian manifolds, Acta Math., Volume 166 (1991), pp. 223-261

[28] Wolf, J.A. 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] Zhou, X.Y. On orbit connectedness, orbit convexity, and envelope of holomorphy, Izv. Russian Akad. Nauk, Ser. Math., Volume 58 (1994) no. 2, pp. 196-205

[30] Zhou, X.Y. On invariant domains in certain complex homogeneous spaces, Ann. Inst. Fourier, Volume 47 (1997) no. 4, pp. 1101-1115

[31] Zhou, X.Y. 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