On the fundamental groups of commutative algebraic groups

Abstract

We say that a smooth algebraic group G over a field k is very special if for any field extension K/k, every G K -homogeneous K-variety has a K-rational point. It is known that every split solvable linear algebraic group is very special. In this note, we show that the converse holds, and discuss its relationship with the birational classification of algebraic group actions.

Homogeneous vector bundles over abelian varieties via representation theory

“…where A is an abelian variety and Y ∈ Pro(L). But this follows from the fact that the pair (Pro(C), Pro(L)) satisfies the lifting property (see [Br18,Cor. 2.12]).…”

“…(i) View G as an extension of an abelian variety A by an affine group scheme H. This readily yields an isomorphism of projective covers P (G) ≃ P (H) ⊕ P (A) with an obvious notation. Moreover, P (H) is affine, and P (A) is an extension of A by an affine group scheme (see [Br18,Prop. 3.3]).…”

“…We now turn to the structure of U A . If p = 0, then U A is the vector group associated with the dual vector space of H 1 (A, O A ) (see [Br18,Lem. 3.8]); thus, U A ≃ (G a ) g . If p > 0, then U A is profinite; more specifically, U A is the largest unipotent quotient of the profinite fundamental group of A, lim ← A[n] (as follows from [Br18, Thm.…”

“…The k-group schemes G A and H A are not of finite type, but their structure is rather well-understood (see [Br18,§3.3] and §2.2). In particular, G A is commutative and geometrically integral; its formation commutes with base change under arbitrary field extensions.…”

Quasi-compact group schemes, Hopf sheaves, and their representations

“…We begin by exhibiting an affine torsor extension G A such that Rep(G A ) ∼ = HVB gr (A). Brion constructs in [12] and [13] the projective cover of A in the category of commutative pro-algebraic group schemes. The corresponding affine torsor extension G A -called the universal extension of the Abelian variety A -is the procured extension.…”