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Complex-symmetric spaces. (English) Zbl 0649.32021
Let X be a compact complex space and G a (not necessarily connected) subgroup of the group of holomorphic automorphisms. Following A. Borel [Arch. Math. 33, 49-56 (1979; Zbl 0423.32015)] we call X complex- symmetric with respect to G, if for \(x\in X\) there exists \(s_ x\in G\) of order two having x as an isolated fixed point. In this paper these spaces are classified under the additional assumption that X is a normal variety and G 0 is reductive (for G 0 semisimple there are partial results of D. N. Akhiezer [Sov. Math., Dokl. 30, 579-582 (1984); translation from Dokl. Akad. Nauk SSSR 279, 13-16 (1984; Zbl 0589.32052)]. It turns out that X is a product of a Hermitian symmetric space and a torus embedding satisfying some additional conditions. In case X is smooth these torus embeddings are classified in detail using the description of torus embeddings by systems of cones and the theory of Coxeter groups. It is most likely that the latter approach also works in the case of singular embeddings.
Reviewer: R.Lehmann

MSC:
32M05 Complex Lie groups, group actions on complex spaces
14L30 Group actions on varieties or schemes (quotients)
14M25 Toric varieties, Newton polyhedra, Okounkov bodies
32M99 Complex spaces with a group of automorphisms
32C15 Complex spaces
32M15 Hermitian symmetric spaces, bounded symmetric domains, Jordan algebras (complex-analytic aspects)
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References:
[1] D. N. AHIEZER, On algebraic varieties that are symmetric in the sense of Borel, Sov. Math. Dokl., 30 (1984), 579-582. · Zbl 0589.32052
[2] A. BIALYNICKI-BIRULA, Some theorems on actions of algebraic groups, Ann. Math., 98 (1973), 480-497. · Zbl 0275.14007
[3] A. BOREL, Symmetric compact complex spaces. Arch. Math., 33 (1979), 49-56. · Zbl 0423.32015
[4] M. BRION, D. LUNA, T. VUST, Espaces homogènes sphériques, Inv. Math., 84 (1986), 617-632. · Zbl 0604.14047
[5] M. BRION, F. PAUER, Valuations des espaces homogènes sphériques, Comm. Math. Helv., 62 (1987), 265-285. · Zbl 0627.14038
[6] E. CARTAN, Œuvres complètes, Gauthier-Villars, Paris, 1952.
[7] H. CARTAN, Quotients of complex analytic spaces, Contributions to Function Theory, Bombay, 1960. · Zbl 0122.08702
[8] H. S. M. COXETER, Discrete groups generated by reflections, Ann. Math., 35 (1934), 588-621. · JFM 60.0898.02
[9] V. I. DANILOV, The geometry of toric varieties, Russian Math. Surveys, 33 (1978), 97-154. · Zbl 0425.14013
[10] A. GROTHENDIECK, J. A. DIEUDONNÉ, Éléments de Géométrie algébrique, Presses Universitaires de France, Paris, 1967.
[11] L. C. GROVE, C. T. BENSON, Finite reflection groups. Graduate Texts in Mathematics, 99, Springer, New York, 1985. · Zbl 0579.20045
[12] R. C. GUNNING, H. ROSSI, Analytic functions of several complex variables, Prentice-Hall, Englewood Cliffs, 1965. · Zbl 0141.08601
[13] S. HELGASON, Differential geometry and symmetric spaces, Academic Press, New York, 1962. · Zbl 0111.18101
[14] H. HOLMANN, Komplexe Räume mit komplexen transformationsgruppen, Math. Ann., 150 (1963), 327-359. · Zbl 0156.30603
[15] A. T. HUCKLEBERRY, E. OELJEKLAUS, Classification theorems for almost homogeneous spaces, Institut Elie Cartan, 9 (1984). · Zbl 0549.32024
[16] K. JÄNICH, Differenzierbare G-mannigfaltigkeiten, Springer, Berlin, 1968. · Zbl 0159.53701
[17] W. KAUP, Bounded symmetric domains in finite and infinite dimensions, Several Complex Variables, Cortona (1976-1977), 180-191. · Zbl 0418.32020
[18] 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. · Zbl 0394.14019
[19] H. KRAFT, Geometrische methoden in der invariantentheorie, Vieweg, Braunschweig, 1984. · Zbl 0569.14003
[20] R. LEHMANN, Complex-symmetric spaces. Dissertation, Bochum, 1988. · Zbl 0689.14025
[21] R. LEHMANN, Complex-symmetric torus embeddings. Schriftenreihe d. Fachbereichs Mathematik der Universität Duisburg, 126 (1987). · Zbl 0697.32015
[22] D. LUNA, T. VUST, Plongements d’espaces homogènes, Comm. Math. Helv., 58 (1983), 186-245. · Zbl 0545.14010
[23] Y. MATSUSHIMA, Fibrés holomorphes sur un tore complexe, Nag. Math. J., 14 (1959), 1-24. · Zbl 0095.36702
[24] D. MONTGOMERY, Simply connected homogeneous spaces, Proc. Am. Math. Soc., 1 (1950), 467-469. · Zbl 0041.36309
[25] G. D. MOSTOW, Self-adjoint groups, Ann. Math., (2), 62 (1955), 44-55. · Zbl 0065.01404
[26] T. ODA, Lectures on torus embeddings and applications. Tata Institute of Fundamental Research, Bombay, 1978. · Zbl 0417.14043
[27] J. POTTERS, On almost homogeneous compact complex surfaces, Inv. Math., 8 (1969), 244-266. · Zbl 0205.25102
[28] H. SUMIHIRO, Equivariant completion. J. Math. Kyoto Univ., 14-1 (1974), 1-28. · Zbl 0277.14008
[29] G. KEMPF, F. KNUDSEN, D. MUMFORD, B. SAINT-DONAT, Toroidal embeddings I, Lecture Notes in Mathematics, 339, Springer, Berlin, 1973. · Zbl 0271.14017
[30] T. VUST, Opération de groupes réductifs dans un type de cônes presque homogènes, Bull. Soc. Math. France, 102 (1974), 317-333. · Zbl 0332.22018
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