Derived categories of Gushel–Mukai varieties. (25th May 2018)

• Record Type: Journal Article
• Authors: Kuznetsov, Alexander; Perry, Alexander
• Is Part Of: Compositio mathematica. Volume 154:Number 7(2018)
• Page Start: 1362
• Page End: 1406
• Publication Date: 2018-05-25
• Subjects: 14F05 (primary), -- 14J45, -- 14E08 (secondary); derived category of coherent sheaves, -- semiorthogonal decomposition, -- K3 categories, -- Fano varieties, -- rationality; Mathematics -- Periodicals; 510
• DOI: 10.1112/S0010437X18007091
• Languages: English
• ISSNs: 0010-437X

Abstract

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 this category is equivalent to the derived category of a K3 surface for a certain codimension 1 family of rational Gushel–Mukai 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.