The group of autoequivalences and the Fourier-Mukai number of a projective manifold

TL;DR

This paper proves that certain smooth projective varieties have no nontrivial Fourier-Mukai partners when their derived autoequivalence group is generated by shifts, automorphisms, and line bundle tensoring.

Contribution

It establishes a criterion linking the structure of the autoequivalence group to the uniqueness of Fourier-Mukai partners for a class of projective varieties.

Findings

No Fourier-Mukai partners exist under the given autoequivalence group conditions.

The autoequivalence group being generated by shifts, automorphisms, and line bundles implies uniqueness.

Provides a new perspective on the relationship between autoequivalence groups and derived equivalences.

Abstract

Let $X$ be a smooth projective variety and $\Aut (D(X))$ the group of autoequivalences of the derived category of $X$. In this paper we show that $X$ has no Fourier-Mukai partner other than $X$ when $\Aut (D(X))$ is generated by shifts, automorphisms and tensor products of line bundles.