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On the geometry of spherical varieties. (English) Zbl 1309.14001
This is a survey article on the theory of spherical varieties which encapsulates in 44 pages (plus 9 pages of bibliographical references) the state of the art of the theory, touching upon essentially all its aspects.
The definition of a spherical variety, with all its equivalent characterizations, is recalled. The Chow group and the homology group of a spherical variety are discussed and compared. The crucial Brion-Luna-Vust Local Structure Theorem is presented, with two of its consequences: rationality of singularities and vanishing theorems.
The Luna-Vust classification of spherical varieties within a given equivariant birational class is illustrated. This allows one to study the divisors of a spherical variety, describing its Picard group, and its ample, nef and globally generated line bundles. Then the class of toroidal spherical varieties is considered, and the fact that every spherical variety has an equivariant toroidal resolution of singularities is explained.
The second special class of spherical varieties which is considered is that of sober spherical varieties, which play a key role in the Luna classification of spherical varieties up to birational equivalence. Their so-called little Weyl group is described and the particular cases of the symmetric varieties and of the wonderful varieties are treated.
The Mori theory for projective spherical varieties is presented. Frobenius splitting techniques to deduce results on spherical varieties in positive characteristic are illustrated. The moment polytope and the Okounkov body associated with a projective spherical variety are described: they encode several geometric properties of the variety itself. The known results about the \(B\)-orbit closures of a spherical variety are presented.

MSC:
14-02 Research exposition (monographs, survey articles) pertaining to algebraic geometry
14M27 Compactifications; symmetric and spherical varieties
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