Geometry of fourfolds with an admissible K3 subcategory 
-- Laura Pertusi, September 21, 2018

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The derived category of a cubic fourfold admits a semiorthogonal decomposition whose non trivial component is a subcategory of K3 type by a result of Kuznetsov. This allowed to prove many properties on the geometry of the hyperkaehler manifolds associated to the cubic fourfold. More recently, Kuznetsov and Perry found a semiorthogonal decomposition with a K3 type component in the case of an other class of fourfolds, known as Gushel-Mukai fourfolds.

The aim of this talk is to discuss a generalization of some results on lattice theory, proved for cubic fourfolds by Addington, Thomas and Huybrechts, in the setting of Gushel-Mukai fourfolds. In particular, we discuss the conditions under which their associated hyperkahler fourfold is birational to a moduli space of stable sheaves (resp. to the Hilbert square) on a K3 surface.