Chow groups and derived categories of K3 surfaces

Summary

The geometry of a K3 surface (over or over) is reflected by its Chow group and its bounded derived category of coherent sheaves in different ways. The Chow group can be infinite dimensional over (Mumford) and is expected to inject into cohomology over (Bloch-Beilinson). The derived category is difficult to describe explicitly, but its group of autoequivalences can be studied by means of the natural representation on cohomology. Conjecturally (Bridgeland) the kernel of this representation is generated by squares of spherical twists. The action of these spherical twists on the Chow ring can be determined explicitly by relating it to the natural subring introduced by Beauville and Voisin.

1. Introduction

In algebraic geometry a K3 surface is a smooth projective surface X over a fixed field K with trivial canonical bundle. For us the field K will be either a number field, the field of algebraic numbers or the complex number field. Nonprojective K3 surfaces play a central role in the theory of K3 surfaces and for some of the results that will be discussed in this text in particular, but here we will not discuss those more analytical aspects.

An explicit example of a K3 surface is provided by the Fermat quartic in given as the zero set of the polynomial. Kummer surfaces, i.e., minimal resolutions of the quotient of abelian surfaces by the sign involution, and elliptic K3 surfaces form other important classes of examples. Most of the results and questions that will be mentioned do not lose any of their interest when considered for one of theses classes of examples or any other particular K3 surface.