Derived categories of Gushel-Mukai varieties

TL;DR

This paper investigates the derived categories of Gushel-Mukai varieties, identifying a special semiorthogonal component that is a K3 or Enriques category, and explores its properties and relations to K3 surfaces and rationality questions.

Contribution

It introduces a new semiorthogonal component in the derived category of Gushel-Mukai varieties and establishes its equivalence to K3 categories in specific cases, advancing understanding of their structure.

Findings

The semiorthogonal component is a K3 or Enriques category depending on dimension.

The K3 category of a Gushel-Mukai fourfold is equivalent to that of a K3 surface in certain cases.

The results support a duality conjecture related to Gushel-Mukai varieties.

Abstract

We study the derived categories of coherent sheaves on Gushel-Mukai varieties. In the derived category of such a variety, we isolate a special semiorthogonal component, which is a K3 or Enriques category according to whether the dimension of the variety is even or odd. We analyze the basic properties of this category using Hochschild homology, Hochschild cohomology, and the Grothendieck group. We study the K3 category of a Gushel-Mukai fourfold in more detail. Namely, we show that this category is equivalent to the derived category of a K3 surface for a certain codimension 1 family of rational fourfolds, and to the K3 category of a birational cubic fourfold for a certain codimension 3 family. The first of these results verifies a special case of a duality conjecture which we formulate. We discuss our results in the context of the rationality problem for Gushel-Mukai varieties, which was one of the main motivations for this work.