Niveau: Supérieur, Doctorat, Bac+8
SOME BASIC RESULTS ON ACTIONS OF NON-AFFINE ALGEBRAIC GROUPS MICHEL BRION Abstract. We study actions of connected algebraic groups on normal algebraic varieties, and show how to reduce them to actions of affine subgroups. 0. Introduction Algebraic group actions have been extensively studied under the as- sumption that the acting group is affine or, equivalently, linear; see [KSS, MFK, Su]. In contrast, little seems to be known about actions of non-affine algebraic groups. In this paper, we show that these actions can be reduced to actions of affine subgroup schemes, in the setting of normal varieties. Our starting point is the following theorem of Nishi and Matsumura (see [Ma]). Let G be a connected algebraic group of automorphisms of a nonsingular algebraic variety X and denote by ?X : X ?? A(X) the Albanese morphism, that is, the universal morphism to an abelian variety (see [Se2]). Then G acts on A(X) by translations, compatibly with its action on X, and the kernel of the induced homomorphism G? A(X) is affine. Applied to the case of G acting on itself via left multiplication, this shows that the Albanese morphism ?G : G ?? A(G) is a surjective group homomorphism having an affine kernel.
- group
- normal gaff -equivariant
- albanese morphism
- any normal
- algebraic group
- variety
- gaff
- acting