Equivalences of derived categories and K3 surfaces

TL;DR

This paper explores the relationships between derived categories of coherent sheaves on smooth projective varieties, establishing conditions for their equivalence, especially focusing on K3 surfaces, and showing how these equivalences can be represented geometrically.

Contribution

It provides a new criterion for when derived categories of two K3 surfaces are equivalent and demonstrates that any derived equivalence can be represented by an object on the product.

Findings

Derived equivalences can be represented by objects on the product variety.

A necessary and sufficient condition for derived equivalence of K3 surfaces is established.

The paper advances understanding of the geometric nature of derived categories.

Abstract

We consider derived categories of coherent sheaves on smooth projective varieties. We prove that any equivalence between them can be represented by an object on the product. Using this, we give a necessary and sufficient condition for equivalence of derived categories of two K3 surfaces.