The K3 category of a cubic fourfold

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.

40 pages, minor corrections, comments on the order of the Brauer classes added, further corrections, to appear in Compositio

References in corpus (7)

• Homological projective duality for Grassmannians of lines
• Hochschild homology and semiorthogonal decompositions
• The Fano variety of lines and rationality problem for a cubic hypersurface
• Derived categories of cubic and V14 threefolds
• A remark on the Torelli theorem for cubic fourfolds
• Integral Transforms and Deformations of K3 Surfaces
• Autoequivalences of twisted K3 surfaces

Cited by in corpus (25)

• Stability conditions on Kuznetsov components
• Derived categories view on rationality problems
• Stability conditions in families
• Derived categories of Gushel-Mukai varieties
• Cubic fourfolds fibered in sextic del Pezzo surfaces
• Stability conditions and moduli spaces for Kuznetsov components of Gushel-Mukai varieties
• Hyperkähler fourfolds and Kummer surfaces
• Birational geometry of the intermediate Jacobian fibration of a cubic fourfold
• Autoequivalences of twisted K3 surfaces
• On the double EPW sextic associated to a Gushel-Mukai fourfold
• Brill-Noether special cubic fourfolds of discriminant 14
• Hochschild cohomology versus the Jacobian ring, and the Torelli theorem for cubic fourfolds
• On the period of Lehn, Lehn, Sorger, and van Straten's symplectic eightfold
• Blowing up linear categories, refinements, and homological projective duality with base locus
• Cubic fourfolds, Kuznetsov components and Chow motives
• Equivalence of K3 surfaces from Verra threefolds
• Singularities of Fano varieties of lines on singular cubic fourfolds
• Hodge numbers and Hodge structures for Calabi-Yau categories of dimension three
• Motives of moduli spaces on K3 surfaces and of special cubic fourfolds
• New rational cubic fourfolds arising from Cremona transformations
• Mirror symmetry and rigid structures of generalized K3 surfaces
• On symplectic birational self-maps of projective hyperkähler manifolds of K3${}^{[n]}$-type
• On lattice polarizable cubic fourfolds
• Derived-natural automorphisms on Hilbert schemes of points on generic K3 surfaces
• The Intermediate Jacobian fibration of a cubic fourfold containing a plane and fibrations in Prym varieties