Rank one connections on abelian varieties, II

TL;DR

This paper constructs an explicit canonical isomorphism between two naturally associated holomorphic torsors over a compact complex torus, enhancing understanding of line bundles and connections on abelian varieties.

Contribution

It provides an explicit construction of a canonical isomorphism between two holomorphic torsors related to line bundles on abelian varieties, extending previous isomorphism results.

Findings

Explicit construction of the isomorphism between the torsors.

Clarification of the relationship between Atiyah sequences and line bundle constructions.

Enhanced understanding of connections on line bundles over abelian varieties.

Abstract

Given a holomorphic line bundle $L$ on a compact complex torus $A$, there are two naturally associated holomorphic $\Omega_A$--torsors over $A$: one is constructed from the Atiyah exact sequence for $L$, and the other is constructed using the line bundle $(p^*_1 L^*)\otimes (\alpha^*L)$, where $\alpha$ is the addition map on $A\times A$, and $p_1$ is the projection of $A\times A$ to the first factor. In \cite{BHR}, it was shown that these two torsors are isomorphic. The aim here is to produce a canonical isomorphism between them through an explicit construction.