The K3 category of a cubic fourfold

TL;DR

This paper explores the deep connections between cubic fourfolds and K3 surfaces, extending the relationship to twisted K3 surfaces and analyzing autoequivalences and period structures.

Contribution

It extends the link between cubic fourfolds and K3 categories to twisted K3 surfaces and determines the autoequivalence group for general cubic fourfolds.

Findings

Determined the autoequivalence group of the K3 category for general cubic fourfolds.

Extended the relationship between cubic fourfolds and K3 surfaces to twisted cases.

Proved finiteness results for cubics with equivalent K3 categories.

Abstract

Smooth cubic fourfolds are linked to K3 surfaces via their Hodge structures, due to work of Hassett, and via Kuznetsov's K3 category A. The relation between these two viewpoints has recently been elucidated by Addington and Thomas. In this paper, both aspects are studied further and extended to twisted K3 surfaces, which in particular allows us to determine the group of autoequivalences of A for the general cubic fourfold. Furthermore, we prove finiteness results for cubics with equivalent K3 categories and study periods of cubics in terms of generalized K3 surfaces.