Derived categories of coherent sheaves on algebraic varieties

Summary

Abstract. In this article, we give the introduction of the recent developments on the derived categories of coherent sheaves on algebraic varieties. We also introduce the notion of stability conditions on triangulated categories in the sense of T. Bridgeland.

Introduction

The notion of derived category of coherent sheaves was first introduced in [24] in order to formulate the Grothendieck duality theorem. It is a category whose objects are complexes of coherent sheaves, and has a structure of a triangulated category. Recently it has been observed that the derived category represents several interesting symmetries, which seems impossible without the notion of derived categories, e.g. McKay correspondence [16], Homological mirror symmetry [43], etc. Now derived categories are a very popular area with interactions with many other subjects including non-commutative algebra, birational geometry, symplectic geometry and string theory. In this article, we give an introduction of the recent results on these topics.

The content of this article is as follows. In Section 2, we give the basic notions concerning derived categories, and propose some fundamental problems. In Section 3, we discuss the relationship between derived category and birational geometry. In Section 4, we discuss the symmetries between derived categories of coherent sheaves and that of module categories of some non-commutative algebras. In Section 5, we introduce the notion of stability conditions on triangulated categories, defined by T. Bridgeland, and see how this notion explains the several symmetries we discuss in this article.