Complex surfaces with equivalent derived categories

TL;DR

This paper investigates how the derived category of coherent sheaves can determine smooth minimal complex projective surfaces, exploring the relationships between different surfaces via Fourier-Mukai transforms.

Contribution

It characterizes the set of surfaces related to a given surface through Fourier-Mukai transforms, advancing understanding of derived categories in algebraic geometry.

Findings

Identifies conditions under which surfaces have equivalent derived categories.

Provides a classification of surfaces related by Fourier-Mukai transforms.

Enhances understanding of the extent to which derived categories determine surface geometry.

Abstract

We examine the extent to which a smooth minimal complex projective surface X is determined by its derived category of coherent sheaves D(X). To do this we find, for each such surface X, the set of surfaces Y for which there exists a Fourier-Mukai transform D(Y) --> D(X).