Dg categories of cubic fourfolds

Original · EN

We prove a reconstruction theorem à la Calabrese-Groechenig for the moduli space parametrizing skyscraper sheaves on a smooth projective variety when these are considered as a system of points in the dg category of perfect complexes on the variety, as axiomatized by Toën and Vaquié. This result is then used to show that, for a cubic fourfold Y⊂ P⁵C, the Kuznetsov category AY is geometric (possibly twisted) if and only if a dg enhancement TY of AY admits a system of points whose associated moduli space is a (possibly twisted) K3 surface.