Fourier-Mukai functors: a survey

TL;DR

This survey reviews recent developments in Fourier--Mukai functors, focusing on their characterization as exact functors between derived categories of coherent sheaves on smooth projective varieties, and discusses open problems in the field.

Contribution

It compiles and discusses recent results on Fourier--Mukai functors, highlighting criteria and open questions about their characterization as derived category functors.

Findings

Several criteria for Fourier--Mukai functors are established

Many open problems regarding the characterization of these functors are presented

The survey summarizes recent progress and unresolved issues in the area

Abstract

This paper surveys some recent results about Fourier--Mukai functors. In particular, given an exact functor between the bounded derived categories of coherent sheaves on two smooth projective varieties, we deal with the question whether this functor is of Fourier--Mukai type. Several related questions are answered and many open problems are stated.