Rank one connections on abelian varieties

TL;DR

This paper studies the structure of the moduli space of rank one algebraic connections on complex abelian varieties, showing how line bundles induce isomorphisms between certain principal bundles and computing their algebraic functions.

Contribution

It establishes an isomorphism between principal bundles associated to line bundles on the dual abelian variety and the moduli space of connections, extending the structure group via a natural homomorphism.

Findings

${ m f C}(L)$ is isomorphic to an extended principal bundle

Computed the algebraic functions on ${ m f C}(L)$

Clarified the relation between line bundles and connection moduli spaces

Abstract

Let A be a complex abelian variety. The moduli space ${\mathcal M}_C$ of rank one algebraic connections on $A$ is a principal bundle over the dual abelian variety $A^\vee=\text{Pic}^0(A)$ for the group $H^0(A, \Omega^1_A)$. Take any line bundle $L$ on $A^\vee$; let ${\mathcal C}(L)$ be the algebraic principal $H^0(A^\vee, \Omega^1_{A^\vee})$-bundle over $A^\vee$ given by the sheaf of connections on $L$. The line bundle $L$ produces a homomorphism $H^0(A, \Omega^1_A) \rightarrow H^0(A^\vee,\, \Omega^1_{A^\vee})$. We prove that ${\mathcal C}(L)$ is isomorphic to the principal $H^0(A^\vee, \Omega^1_{A^\vee})$-bundle obtained by extending the structure group of the principal $H^0(A,\, \Omega^1_A)$-bundle ${\mathcal M}_C$ using this homomorphism given by $L$. We compute the ring of algebraic functions on ${\mathcal C}(L)$.