Homogeneous vector bundles over abelian varieties via representation theory

Abstract

Let A be an abelian variety over a field. The homogeneous (or translation-invariant) vector bundles over A form an abelian category HVec A ; the Fourier-Mukai transform yields an equivalence of HVec A with the category of coherent sheaves with finite support on the dual abelian variety. In this paper, we develop an alternative approach to homogeneous vector bundles, based on the equivalence of HVec A with the category of finite-dimensional representations of a commutative affine group scheme (the "affine fundamental group" of A). This displays remarkable analogies between homogeneous vector bundles over abelian varieties and representations of split reductive algebraic groups.

Irreducible modules for pseudo-reductive groups

“…/ naturally identifies as a subfield of the field of definition of the unipotent radical of C . Similar results have been proved by Brion [1,Prop. 3.1] in a slightly different context, and using a different approach.…”

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.…”

“…On the other hand, the construction of Aut ⊗ (Id) shows that G A is the limit of the system of Aut gr (E), when we consider the order: Aut gr (E ′ ) ≥ Aut gr (E) if and only if there exists E ′′ ∈ HVB gr (A) such that E = E ′ ⊕ E ′′ . We obtain in this way the description of the universal extension of the Abelian variety A as the limit of the family of automorphism groups of homogeneous vector bundles, as in [13].…”

“…In particular, the affine torsor extension S is exact if and only if given an affine scheme S = Spec(B) and a T -point a : T → A ∈ A(T ), then there exist an affine scheme T ′ = Spec(B ′ ), faithfully flat of finite presentation over T , and a T ′ -point g : 13. Let H be a Hopf sheaf.…”

On the fundamental groups of commutative algebraic groups

“…As an application of the above developments, we obtain a spectral sequence à la Milne (see [Mi70]), which relates the extension groups in C and in the corresponding category over a Galois extension of k. Further applications, to the structure of homogeneous vector bundles over abelian varieties, are presented in [Br18].…”