Complete homogeneous varieties: Structure and classification

Abstract

Abstract. Homogeneous varieties are those whose group of automorphisms acts transitively on them. In this paper we prove that any complete homogeneous variety splits in a unique way as a product of an abelian variety and a parabolic variety. This is obtained by proving a rigidity theorem for the parabolic subgroups of a linear group. Finally, using the results of Wenzel on the classification of parabolic subgroups of a linear group and the results of Demazure on the automorphisms of a flag variety, we obtain the classification of the parabolic varieties (in characteristic different from 2, 3). This, together with the moduli of abelian varieties, concludes the classification of the complete homogeneous varieties. IntroductionLet X be a variety over an algebraically closed field k. Let G be the functor of automorphisms of X that acts naturally on X. We say that X is homogeneous if G acts transitively on X, that is, if for each pair of points x, x ∈ Hom(S, X), there exists an automorphism τ ∈ Aut S (X × S) (after a faithfully flat base change on S) transforming one into the other:It is known that the group of automorphisms of a complete variety exists, i.e., the functor G is representable, and it is locally of finite type (see [7]). Hence, if X is smooth, connected and homogeneous, then the reduced connected component through the origin of G acts transitively on X. Therefore, every smooth and connected homogeneous variety is isomorphic to G/P , with G a smooth and connected algebraic group and P ⊂ G a subgroup.Let Aut 0 (X) denote the reduced connected component through the origin of the automorphism scheme of X.The main results that we obtain here are:(1) A complete homogeneous variety splits canonically and uniquely as a direct product of an abelian variety and a parabolic variety.A parabolic variety means a variety of the form G/P with G an affine, smooth and connected algebraic group and P a parabolic subgroup (eventually not reduced), i.e., a subgroup containing a Borel subgroup of G. This is Theorem 5.2. The analogue of this result for compact Kähler manifolds is due to A. Borel and R. Remmer (see [2]).

Motivic Decomposition of Projective Pseudo-Homogeneous Varieties

“…In [37], Wenzel has classified all parabolic subgroup schemes P and in [38] he proved that the varieties G P are rational. Using this classification, de Salas in [34] has classified all G P . The varieties of the form G P where P is any parabolic subgroup scheme that may or may not be reduced are known as parabolic varieties in [34].…”

“…Using this classification, de Salas in [34] has classified all G P . The varieties of the form G P where P is any parabolic subgroup scheme that may or may not be reduced are known as parabolic varieties in [34]. Lauritzen and Haboush answered many interesting questions about the geometry of these varieties including canonical line bundles, vanishing theorems and Frobenius splitting in [30], [17] and [28].…”

D-modules and projective stacks

“…Recall that a variety X is homogeneous if a connected algebraic (not necessarily linear) group G acts transitively on X. For a complete variety X it is equivalent to asking that the automorphism group of X acts transitively on X [13]. Such X is necessarily smooth.…”

“…Proof. By Borel-Remmert Theorem [13] X is a product of a partial flag variety and an abelian variety A. It remains to notice that A is not D-affine because R dimA Γ(A, O A ) = 0 by Serre's duality, unless A is a point.…”

Principal bundles, quasi-abelian varieties and structure of algebraic groups

“…A second motivation comes from the problem of classification of homogeneous varieties. This problem is essentially solved in the proper case (see [Sa03]). The next step is to deal with the anti-affine case (anti-affine means that the variety has only constant global functions).…”