Some finiteness results for Fourier-Mukai partners

TL;DR

This paper develops methods to study Fourier-Mukai partners of algebraic varieties, proving finiteness for abelian varieties and their unique determination by derived categories, while generalizing a key theorem.

Contribution

It introduces new techniques for analyzing Fourier-Mukai partners and extends a significant theorem by Bondal and Orlov.

Findings

Abelian varieties have finitely many Fourier-Mukai partners.

Fourier-Mukai partners are uniquely determined by their derived category of coherent D-modules.

Generalization of Bondal and Orlov's theorem.

Abstract

We develop some methods for studying the Fourier-Mukai partners of an algebraic variety. As applications we prove that abelian varieties have finitely many Fourier-Mukai partners and that they are uniquely determined by their derived category of coherent $D$-modules. We also generalize a famous theorem due to A. Bondal and D. Orlov.