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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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