Fully faithful Fourier-Mukai functors and generic vanishing

TL;DR

This paper shows that the condition for a Fourier-Mukai functor to be fully faithful can be understood through the lens of generic vanishing, connecting classical examples with this conceptual framework.

Contribution

It reveals that the fully faithfulness of Fourier-Mukai functors is an instance of generic vanishing, providing a new perspective on classical criteria.

Findings

Fully faithfulness corresponds to generic vanishing conditions.

Classical examples like Bondal-Orlov criterion are explained via this perspective.

Connections are made with standard flip and Mukai flop cases.

Abstract

The aim of this mainly expository note is to point out that, given an Fourier-Mukai functor, the condition making it fully faithful is an instance of \emph{generic vanishing}. We test this point of view on some fairly classical examples, including the strong simplicity criterion of Bondal and Orlov, the standard flip and the Mukai flop.