Derived categories of Gushel-Mukai varieties. (English) Zbl 1401.14181
Working over an algebraically closed field of characteristic 0, the authors show that the derived category of a Gushel-Mukai (GM) variety admits a semiorthogonal decomposition with a GM category as its component. This GM category is proved to be a \(K3\) or Enriques category depending on the parity of the dimension of the Gushel-Mukai variety. They compute the Hochschild homology, Hochschild cohomology and the numerical Grothendieck group of this GM category. These computations allow them to conclude that for any Gushel-Mukai variety of odd dimension or a very general Gushel Mukai variety of even dimension greater than 2, the GM category is not equivalent to the derived category of any variety. They introduce the notions of generalized dual GM varieties and generalized GM partners, and conjecture that they have equivalent GM categories. They show this conjecture in a special case for a certain codimension 1 family of GM fourfolds. They also conjecture that GM category of a rational GM fourfold is equivalent to the derived category of a \(K3\) surface. This conjecture together with their result on the non-geometricity of the GM category for a very general GM fourfold, is the first conjectural example of an irrational GM fourfold. Lastly, in the appendix they prove that the GM varieties of a fixed dimension form a smooth and irreducible Deligne-Mumford stack and they compute the dimension of this stack.
Reviewer: Tanya Srivastava (Klosterneuburg)
Keywords:
derived categories of coherent sheaves; Gushel-Mukai varieties; \(K3\) category; rationality; cubic fourfoldsReferences:
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