Abstract
We consider complex manifolds that admit actions by holomorphic transformations of classical simple real Lie groups and classify all such manifolds in a natural situation. Under our assumptions, which require the group at hand to be dimension-theoretically large with respect to the manifold on which it is acting, our classification result states that the manifolds which arise are described precisely as invariant open subsets of certain complex flag manifolds associated to the complexified groups.
Similar content being viewed by others
References
Altomani, A., Medori, C., Nacinovich, M.: Orbits of real groups in complex flag manifolds. Preprint, available from http://front.math.ucdavis.edu/0711.4484
Barut, A., Bracken, A.: The remarkable algebra \({\mathfrak{so}}^{*}(2n)\) , its representations, its Clifford algebra and potential applications. J. Phys. A Math. Gen. 23, 641–663 (1990)
Byun, J., Kodama, A., Shimizu, S.: A group-theoretic characterization of the direct product of a ball and a Euclidean space. Forum Math. 18, 983–1009 (2006)
Darmstadt, S.: On complex Lie algebras with a simple real form. Semin. Sophus Lie 1, 243–245 (1991). Seminar Sophus Lie (Darmstadt, 1991)
Djoković, D.: On real forms of complex semisimple Lie algebras. Aeq. Math. 58, 73–84 (1999)
Fels, G., Huckleberry, A., Wolf, J.: Cycle Spaces of Flag Domains. A Complex Geometric Viewpoint. Progress in Mathematics, vol. 245. Birkhäuser, Basel (2006)
Helgason, S.: Differential Geometry and Symmetric Spaces. Academic Press, San Diego (1962)
Hochschild, G.: Complexification of real analytic groups. Trans. Am. Math. Soc. 125, 406–413 (1966)
Huckleberry, A.T.: Actions of groups of holomorphic transformations. In: Several Complex Variables, VI. Encyclopedia Math. Sci., vol. 69, pp. 143–196. Springer, Berlin (1990)
Huckleberry, A.T., Isaev, A.V.: Classical symmetries of complex manifolds. Preprint, available from http://arxiv.org/abs/0901.4280
Huckleberry, A., Oeljeklaus, E.: Classification Theorems for Almost Homogeneous Spaces. Revue de l’Institut Elie Cartan, vol. 9. University of Nancy, Nancy (1984)
Isaev, A.V.: Characterization of the unit ball in ℂn among complex manifolds of dimension n. J. Geom. Anal. 14, 697–700 (2004). Erratum: J. Geom. Anal. 18, 919 (2008)
Isaev, A.V.: Hyperbolic manifolds of dimension n with automorphism group of dimension n 2−1. J. Geom. Anal. 15, 239–259 (2005)
Isaev, A.V.: Hyperbolic manifolds with high-dimensional automorphism group. Proc. Steklov Inst. Math. 253, 225–245 (2006). Collected Papers in the Memory of A.G. Vitushkin
Isaev, A.V.: Lectures on the Automorphism Groups of Kobayashi-Hyperbolic Manifolds. Lecture Notes in Mathematics, vol. 1902. Springer, Berlin (2007)
Isaev, A.V.: Hyperbolic 2-dimensional manifolds with 3-dimensional automorphism group. Geom. Topol. 12, 643–711 (2008)
Isaev, A.V., Kruzhilin, N.G.: Effective actions of the unitary group on complex manifolds. Can. J. Math. 54, 1254–1279 (2002)
Isaev, A.V., Kruzhilin, N.G.: Effective actions of SU n on complex n-dimensional manifolds. Ill. J. Math. 48, 37–57 (2004)
Onishchik, A.L.: Topology of Transitive Transformation Groups. Barth, Leipzig (1994)
Onishchik, A., Vinberg, E.: Lie Groups and Algebraic Groups. Springer, Berlin (1990)
Snow, D., Winkelmann, J.: Compact complex homogeneous manifolds with large automorphism groups. Invent. Math. 134, 139–144 (1998)
Stuck, G.: Low dimensional actions of semisimple groups. Isr. J. Math. 76, 27–71 (1991)
Tomuro, T.: Smooth SL(n,ℍ), Sp(n,ℂ)-actions on (4n−1)-manifolds. Tohôku Math. J. 44(2), 243–250 (1992)
Uchida, F.: Smooth actions of special unitary groups on cohomology complex projective spaces. Osaka J. Math. 12, 375–400 (1975)
Uchida, F.: Smooth SL(n,ℂ) actions on (2n−1)-manifolds. Hokkaido Math. J. 21, 79–86 (1992)
Uchida, F., Mukōyama, K.: Smooth actions of non-compact semi-simple Lie groups. In: Current Trends in Transformation Groups. K-Monographs in Mathematics, vol. 7, pp. 201–215. Kluwer Academic, Dordrecht (2002)
Winkelmann, J.: The Classification of Three-Dimensional Homogeneous Complex Manifolds. Lecture Notes in Mathematics, vol. 1602. Springer, Berlin (1995)
Wolf, J.: The action of a real semisimple group on a complex flag manifold. I. Orbit structure and holomorphic arc components. Bull. Am. Math. Soc. 75, 1121–1237 (1969)
Wolf, J.: Real groups transitive on complex flag manifolds. Proc. Am. math. Soc. 129, 2483–2487 (2001)
Zierau, R.: Representations in Dolbeault cohomology. In: Representation Theory of Lie Groups. IAS/Park City Mathematics Series, vol. 8, pp. 89–146. Am. Math. Soc., Providence (2000)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Kang-Tae Kim.
Rights and permissions
About this article
Cite this article
Huckleberry, A., Isaev, A. Classical Symmetries of Complex Manifolds. J Geom Anal 20, 132–152 (2010). https://doi.org/10.1007/s12220-009-9095-6
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12220-009-9095-6