Spherical Varieties

Abstract

Although the theory developed in the previous chapters applies to arbitrary homogeneous spaces of reductive groups, and even to more general group actions, it acquires its most complete and elegant form for spherical homogeneous spaces and their equivariant embeddings, called spherical varieties. A justification of the fact that spherical homogeneous spaces are a significant mathematical object is that they arise naturally in various fields, such as embedding theory, representation theory, symplectic geometry, etc. In §25 we collect various characterizations of spherical spaces, the most important being: the existence of an open B-orbit, the “multiplicity-free” property for spaces of global sections of line bundles, commutativity of invariant differential operators and of invariant functions on the cotangent bundle with respect to the Poisson bracket.

Then we examine the most interesting classes of spherical homogeneous spaces and spherical varieties in more detail. Algebraic symmetric spaces are considered in §26. We develop the structure theory and classification of symmetric spaces, compute the colored data required for the description of their equivariant embeddings, and study B-orbits and (co)isotropy representation. §27 is devoted to (G×G)-equivariant embeddings of a reductive group G. A particular interest in this class is explained, for example, by an observation that linear algebraic monoids are nothing else but affine equivariant group embeddings. Horospherical varieties of complexity 0 are classified and studied in §28.

The geometric structure of toroidal varieties, considered in §29, is the best understood among all spherical varieties, since toroidal varieties are “locally toric”. They can be defined by several equivalent properties: their fans are “colorless”, they are spherical and pseudo-free, and the action sheaf on a toroidal variety is the log-tangent sheaf with respect to a G-stable divisor with normal crossings. An important property of toroidal varieties is that they are rigid as G-varieties. The so-called wonderful varieties are the most remarkable subclass of toroidal varieties. They are canonical completions with nice geometric properties of (certain) spherical homogeneous spaces. The theory of wonderful varieties is developed in §30. Applications include computation of the canonical divisor of a spherical variety and Luna’s conceptual approach to the classification of spherical subgroups through the classification of wonderful varieties.

The concluding §31 is devoted to Frobenius splitting, a technique for proving geometric and algebraic properties (normality, rationality of singularities, cohomology vanishing, etc) in positive characteristic. However, this technique can be applied to zero characteristic using reduction mod p provided that reduced varieties are Frobenius split. This works for spherical varieties. As a consequence, one obtains the vanishing of higher cohomology of ample or numerically effective line bundles on complete spherical varieties, normality and rationality of singularities for G-stable subvarieties, etc. Some of these results can be proved by other methods, but Frobenius splitting provides a simple uniform approach.