General Theory of Embeddings

Abstract

Equivariant embeddings of homogeneous spaces are one of the main topics of this survey. The general theory of them was developed by D. Luna and Th. Vust in a fundamental paper [Plongements d’espaces homogènes, Comment. Math. Helv. 58 (1983), no. 2, 186–245]. However it was noticed in [D. A. Timashev, Classification of G-varieties of complexity 1, Math. USSR-Izv. 61 (1997), no. 2, 363–397] that the whole theory admits a natural exposition in a more general framework, which is discussed in this chapter. The generically transitive case differs from the general one by the existence of a smallest G-variety of a given birational type, namely, a homogeneous space.

In §12 we discuss the general approach of Luna and Vust based on patching all G-varieties of a given birational class together in one huge prevariety and studying particular G-varieties as open subsets in it. An important notion of a B-chart arising in such a local study is considered in §13. A B-chart is a B-stable affine open subset of a G-variety, and any normal G-variety is covered by (finitely many) G-translates of B-charts. B-charts and their “admissible” collections corresponding to coverings of G-varieties are described in terms of colored data composed of B-stable divisors and G-invariant valuations of a given function field. This leads to a “combinatorial” description of normal G-varieties in terms of colored data, obtained in §14. In the cases of complexity ≤1, considered in §§15–16, this description is indeed combinatorial, namely, in terms of polyhedral cones, their faces, fans, and other objects of combinatorial convex geometry.

Divisors on G-varieties are studied in §17. We give criteria for a divisor to be Cartier, finitely generated and ample, and we describe global sections in terms of colored data. Aspects of the intersection theory on a G-variety are discussed in §18, including the rôle of B-stable cycles and a formula for the degree of an ample divisor.