Complex-symmetric spaces
Lehmann, Ralf
Annales de l'Institut Fourier, Tome 39 (1989), p. 373-416 / Harvested from Numdam
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Un espace compact complexe X est appelé complexe-symétrique relatif à un sous-groupe G du groupe Aut θ (X) si chaque point x∈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 Aut 0 (X), 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.

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     author = {Lehmann, Ralf},
     title = {Complex-symmetric spaces},
     journal = {Annales de l'Institut Fourier},
     volume = {39},
     year = {1989},
     pages = {373-416},
     doi = {10.5802/aif.1171},
     mrnumber = {91d:32048},
     zbl = {0649.32021},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIF_1989__39_2_373_0}
}

Lehmann, Ralf. Complex-symmetric spaces. Annales de l'Institut Fourier, Tome 39 (1989) pp. 373-416. doi : 10.5802/aif.1171. http://gdmltest.u-ga.fr/item/AIF_1989__39_2_373_0/

[A] D. N. Ahiezer, On Algebraic Varieties that are Symmetric in the Sense of Borel, Sov. Math. Dokl., 30 (1984), 579-582. | Zbl 0589.32052

[BB] A. Bialynicki-Birula, Some Theorems on Actions of Algebraic Groups, Ann. Math., 98 (1973), 480-497. | MR 51 #3186 | Zbl 0275.14007

[Bo] A. Borel, Symmetric Compact Complex Spaces. Arch. Math., 33 (1979), 49-56. | MR 80k:32033 | Zbl 0423.32015

[B-L-V] M. Brion, D. Luna, T. Vust, Espaces homogènes sphériques, Inv. Math., 84 (1986), 617-632. | MR 87g:14057 | Zbl 0604.14047

[B-P] M. Brion, F. Pauer, Valuations des espaces homogènes sphériques, Comm. Math. Helv., 62 (1987), 265-285. | MR 88h:14051 | Zbl 0627.14038

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

[Car] H. Cartan, Quotients of Complex Analytic Spaces, Contributions to Function Theory, Bombay, 1960. | MR 25 #3199 | Zbl 0122.08702

[Cox] H. S. M. Coxeter, Discrete Groups Generated by Reflections, Ann. Math., 35 (1934), 588-621. | JFM 60.0898.02 | Zbl 0010.01101

[Dan] V. I. Danilov, The Geometry of Toric Varieties, Russian Math. Surveys, 33 (1978), 97-154. | MR 80g:14001 | Zbl 0425.14013

[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 85m:20001 | Zbl 0579.20045

[G-R] R. C. Gunning, H. Rossi, Analytic Functions of Several Complex Variables, Prentice-Hall, Englewood Cliffs, 1965. | MR 31 #4927 | Zbl 0141.08601

[Hel] S. Helgason, Differential Geometry and Symmetric Spaces, Academic Press, New York, 1962. | MR 26 #2986 | Zbl 0111.18101

[Ho] H. Holmann, Komplexe Räume mit komplexen Transformationsgruppen, Math. Ann., 150 (1963), 327-359. | MR 27 #776 | Zbl 0156.30603

[H-O] A. T. Huckleberry, E. Oeljeklaus, Classification Theorems for Almost Homogeneous Spaces, Institut Elie Cartan, 9 (1984). | MR 86g:32050 | Zbl 0549.32024

[Jän] K. Jänich, Differenzierbare G-Mannigfaltigkeiten, Springer, Berlin, 1968. | Zbl 0159.53701

[K] W. Kaup, Bounded Symmetric Domains in Finite and Infinite Dimensions, Several Complex Variables, Cortona (1976-1977), 180-191. | MR 84b:32045 | Zbl 0418.32020

[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 81b:14026 | Zbl 0394.14019

[Kr] H. Kraft, Geometrische Methoden in der Invariantentheorie, Vieweg, Braunschweig, 1984. | MR 86j:14006 | Zbl 0569.14003

[L1] R. Lehmann, Complex-symmetric Spaces. Dissertation, Bochum, 1988. | Zbl 0689.14025

[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 85a:14035 | Zbl 0545.14010

[Ma] Y. Matsushima, Fibrés holomorphes sur un tore complexe, Nag. Math. J., 14 (1959), 1-24. | MR 21 #1403 | Zbl 0095.36702

[Mon] D. Montgomery, Simply Connected Homogeneous Spaces, Proc. Am. Math. Soc., 1 (1950), 467-469. | MR 12,242c | Zbl 0041.36309

[Mo] G. D. Mostow, Self-adjoint Groups, Ann. Math., (2), 62 (1955), 44-55. | MR 16,1088a | Zbl 0065.01404

[Oda] T. Oda, Lectures on Torus Embeddings and Applications. Tata Institute of Fundamental Research, Bombay, 1978. | MR 81e:14001 | Zbl 0417.14043

[P] J. Potters, On Almost Homogeneous Compact Complex Surfaces, Inv. Math., 8 (1969), 244-266. | MR 41 #3808 | Zbl 0205.25102

[Su] H. Sumihiro, Equivariant Completion. J. Math. Kyoto Univ., 14-1 (1974), 1-28. | MR 49 #2732 | Zbl 0277.14008

[T-E] G. Kempf, F. Knudsen, D. Mumford, B. Saint-Donat, Toroidal Embeddings I, Lecture Notes in Mathematics, 339, Springer, Berlin, 1973. | MR 49 #299 | Zbl 0271.14017

[V] T. Vust, Opération de groupes réductifs dans un type de cônes presque homogènes, Bull. Soc. Math. France, 102 (1974), 317-333. | Numdam | MR 51 #3187 | Zbl 0332.22018