Derived categories of Gushel–Mukai varieties

By Alexander Kuznetsov, Alexander Perry. Compositio Mathematica, 2016

84 citations recorded by the source corpus.

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.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/020c30c840d8d27c8287e5d7bb2f031b104d0967

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Geometry and Topology

1. P. Stellari
2. Emanuele Macrì
3. Alexander Perry
4. D. Huybrechts
5. Arend Bayer
6. Chunyi Li
7. Alexander Kuznetsov
8. Benjamin Schmidt

In Mathematical Physics

1. P. Stellari
2. Emanuele Macrì
3. Alexander Perry
4. D. Huybrechts
5. Arend Bayer
6. Alexander Kuznetsov
7. Chunyi Li
8. Benjamin Schmidt

Citations in the corpus

• The integral Hodge conjecture for two-dimensional Calabi–Yau categories (2020, background): “In particular, by [57] for any GM variety there is a Kuznetsov component Ku(X) ⊂ Dperf(X) defined by the semiorthogonal decomposition Dperf(X) = 〈 Ku(X),OX ,U ∨ X , .”
• Lectures on Non-commutative K3 Surfaces, Bridgeland Stability, and Moduli Spaces (2018, background): “ will only shortly recall the definition of Kuznetsov component for n= 4,6, and mention a few results towards Question 3.2 and Question 3.3 in these cases, mostly without proofs. Our main reference is [KuPe18a], and we refer there for all details. The Kuznetsov component. We assume n= 4,6.6 In this c”