Complex-symmetric spaces

Lehmann, Ralf

Annales de l'Institut Fourier, Tome 39 (1989) no. 2, pp. 373-416.

Un espace compact complexe $X$ est appelé complexe-symétrique relatif à un sous-groupe $G$ du groupe ${\mathrm{Aut}}_{\theta }\left(X\right)$ si chaque point $x\in X$ est un point fixe isolé d’un automorphisme involutif dans $G$. Il en résulte que $X$ est presque $G^0$-homogène. Après quelques exemples nous classifions les variétés complexes-symétriques normales avec $G^0$ réductif. Nous prouvons que $X$ est un produit d’un espace hermitien symétrique et d’un plongement d’un tore algébrique satisfaisant quelques conditions supplémentaires. Dans le cas lisse ces plongements sont classifiés en utilisant la description d’un plongement par un système de cônes (éventails) et la théorie de groupes de Coxeter.

A compact complex space $X$ is called complex-symmetric with respect to a subgroup $G$ of the group ${\mathrm{Aut}}_0\left(X\right)$, if each point of $X$ is isolated fixed point of an involutive automorphism of $G$. It follows that $G$ is almost $G^0$-homogeneous. After some examples we classify normal complex-symmetric varieties with $G^0$ reductive. It turns out that $X$ is a product of a Hermitian symmetric space and a compact torus embedding satisfying some additional conditions. In the smooth case these torus embeddings are classified using the description of torus embeddings by systems of cone (“fans”) and the theory of Coxeter groups.

[A] D. N. Ahiezer, On Algebraic Varieties that are Symmetric in the Sense of Borel, Sov. Math. Dokl., 30 (1984), 579-582. | Zbl

[BB] A. Bialynicki-Birula, Some Theorems on Actions of Algebraic Groups, Ann. Math., 98 (1973), 480-497. | MR | Zbl

[Bo] A. Borel, Symmetric Compact Complex Spaces. Arch. Math., 33 (1979), 49-56. | MR | Zbl

[B-L-V] M. Brion, D. Luna, T. Vust, Espaces homogènes sphériques, Inv. Math., 84 (1986), 617-632. | MR | Zbl

[B-P] M. Brion, F. Pauer, Valuations des espaces homogènes sphériques, Comm. Math. Helv., 62 (1987), 265-285. | MR | Zbl

[Ca] E. Cartan, Œuvres Complètes, Gauthier-Villars, Paris, 1952.

[Car] H. Cartan, Quotients of Complex Analytic Spaces, Contributions to Function Theory, Bombay, 1960. | MR | Zbl

[Cox] H. S. M. Coxeter, Discrete Groups Generated by Reflections, Ann. Math., 35 (1934), 588-621. | JFM | Zbl

[Dan] V. I. Danilov, The Geometry of Toric Varieties, Russian Math. Surveys, 33 (1978), 97-154. | MR | Zbl

[EGA] A. Grothendieck, J. A. Dieudonné, Éléments de Géométrie Algébrique, Presses Universitaires de France, Paris, 1967. | Numdam

[G-B] L. C. Grove, C. T. Benson, Finite Reflection Groups. Graduate Texts in Mathematics, 99, Springer, New York, 1985. | MR | Zbl

[G-R] R. C. Gunning, H. Rossi, Analytic Functions of Several Complex Variables, Prentice-Hall, Englewood Cliffs, 1965. | MR | Zbl

[Hel] S. Helgason, Differential Geometry and Symmetric Spaces, Academic Press, New York, 1962. | MR | Zbl

[Ho] H. Holmann, Komplexe Räume mit komplexen Transformationsgruppen, Math. Ann., 150 (1963), 327-359. | MR | Zbl

[H-O] A. T. Huckleberry, E. Oeljeklaus, Classification Theorems for Almost Homogeneous Spaces, Institut Elie Cartan, 9 (1984). | MR | Zbl

[Jän] K. Jänich, Differenzierbare G-Mannigfaltigkeiten, Springer, Berlin, 1968. | Zbl

[K] W. Kaup, Bounded Symmetric Domains in Finite and Infinite Dimensions, Several Complex Variables, Cortona (1976-1977), 180-191. | MR | Zbl

[Ko] J. Konarski, Decompositions of Normal Algebraic Varieties Determined by an Action of a One-dimensional Torus, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys., 26 (1978), 295-300. | MR | Zbl

[Kr] H. Kraft, Geometrische Methoden in der Invariantentheorie, Vieweg, Braunschweig, 1984. | MR | Zbl

[L1] R. Lehmann, Complex-symmetric Spaces. Dissertation, Bochum, 1988. | Zbl

[L2] R. Lehmann, Complex-symmetric Torus Embeddings. Schriftenreihe d. Fachbereichs Mathematik der Universität Duisburg, 126 (1987).

[L-V] D. Luna, T. Vust, Plongements d'espaces homogènes, Comm. Math. Helv., 58 (1983), 186-245. | MR | Zbl

[Ma] Y. Matsushima, Fibrés holomorphes sur un tore complexe, Nag. Math. J., 14 (1959), 1-24. | MR | Zbl

[Mon] D. Montgomery, Simply Connected Homogeneous Spaces, Proc. Am. Math. Soc., 1 (1950), 467-469. | MR | Zbl

[Mo] G. D. Mostow, Self-adjoint Groups, Ann. Math., (2), 62 (1955), 44-55. | MR | Zbl

[Oda] T. Oda, Lectures on Torus Embeddings and Applications. Tata Institute of Fundamental Research, Bombay, 1978. | MR | Zbl

[P] J. Potters, On Almost Homogeneous Compact Complex Surfaces, Inv. Math., 8 (1969), 244-266. | MR | Zbl

[Su] H. Sumihiro, Equivariant Completion. J. Math. Kyoto Univ., 14-1 (1974), 1-28. | MR | Zbl

[T-E] G. Kempf, F. Knudsen, D. Mumford, B. Saint-Donat, Toroidal Embeddings I, Lecture Notes in Mathematics, 339, Springer, Berlin, 1973. | MR | Zbl

[V] T. Vust, Opération de groupes réductifs dans un type de cônes presque homogènes, Bull. Soc. Math. France, 102 (1974), 317-333. | Numdam | MR | Zbl