Fourier-Mukai transforms of line bundles on derived equivalent abelian varieties

TL;DR

This paper investigates the properties of Fourier-Mukai transforms of line bundles on derived equivalent abelian varieties, revealing a link between ampleness and nefness of parametrized bundles.

Contribution

It establishes a criterion connecting the ampleness of Fourier-Mukai transforms of negative line bundles to the nefness of the parametrized bundles on abelian varieties.

Findings

Fourier-Mukai transform of a very negative line bundle is ample if and only if the bundles are nef.

Provides a characterization of when the Fourier-Mukai transform yields ample sheaves.

Links geometric properties of line bundles to the positivity of associated vector bundles.

Abstract

We study the Fourier-Mukai functor D(Y) -> D(X) induced by the universal family on a fine moduli space Y for simple semihomogeneous vector bundles on an abelian variety X. The main result is that the Fourier-Mukai transform of a very negative line bundle on Y is ample if and only if the bundles parametrized by Y are nef.