Cohomology Groups of Deformations of Line Bundles on Complex Tori

TL;DR

This paper computes the cohomology groups of line bundles on complex tori under various holomorphic, non-commutative deformations, bridging classical and non-commutative cases.

Contribution

It introduces methods to analyze cohomology groups of line bundles across both commutative and non-commutative deformations of complex tori.

Findings

Cohomology groups for line bundles over constant, commutative deformations are computed.

Cohomology groups for non-commutative deformations of degree-zero line bundles are determined.

The analysis interpolates between classical and non-commutative deformation cases.

Abstract

The cohomology groups of line bundles over complex tori (or abelian varieties) are classically studied invariants of these spaces. In this article, we compute the cohomology groups of line bundles over various holomorphic, non-commutative deformations of complex tori. Our analysis interpolates between two extreme cases. The first case is a calculation of the space of (cohomological) theta functions for line bundles over constant, commutative deformations. The second case is a calculation of the cohomologies of non-commutative deformations of degree-zero line bundles.