Bundles of generalized theta functions over abelian surfaces

TL;DR

This paper investigates bundles of generalized theta functions on abelian surfaces, revealing their splitting types, symmetries, and transformation formulas, thus extending known results from curves to higher-dimensional surfaces.

Contribution

It introduces a new analysis of Verlinde bundles on abelian surfaces, including their splitting, symmetries, and a generalized transformation formula for theta bundles.

Findings

Splitting type expressed via indecomposable semihomogeneous factors

Fourier-Mukai symmetries of Verlinde bundles identified

Derived a transformation formula extending Drezet-Narasimhan theorem

Abstract

We study the bundles of generalized theta functions constructed from moduli spaces of sheaves over abelian surfaces. In degree 0, the splitting type of these bundles is expressed in terms of indecomposable semihomogeneous factors. Furthermore, Fourier-Mukai symmetries of the Verlinde bundles are found, consistently with strange duality. Along the way, a transformation formula for the theta bundles is derived, extending a theorem of Drezet-Narasimhan from curves to abelian surfaces.