We classify principal bundles over anti-affine schemes with affine and commutative structure group. We show that this yields the classification of quasi-abelian varieties over a field k (i.e., group k-schemes G such that O G (G) = k). The interest of this result is given by the fact that the classification of smooth group k-schemes is reduced to the classification of quasi-abelian varieties and of certain affine group schemes.

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]).

Abstract. The supersymmetric product of a supercurve is constructed with the aid of a theorem of algebraic invariants and the notion of positive relative superdivisor (supervortex) is introduced. A supercurve of positive superdivisors of degree 1 (supervortices of vortex number 1) of the original supercurve is constructed as its supercurve of conjugate fermions, as well as the supervariety of relative positive superdivisors of degre p (supervortices of vortex number p.) A universal superdivisor is defined and it is proved that every positive relative superdivisor can be obtained in a unique way as a pull-back of the universal superdivisor. The case of SUSY-curves is discussed.

No abstract

No abstract