On the symplectic eightfold associated to a Pfaffian cubic fourfold

Abstract

We show that the irreducible holomorphic symplectic eightfold Z associated to a cubic fourfold Y not containing a plane is deformation-equivalent to the Hilbert scheme of four points on a K3 surface. We do this by constructing for a generic Pfaffian cubic Y a birational map Z ---> Hilb^4(X), where X is the K3 surface associated to Y by Beauville and Donagi. We interpret Z as a moduli space of complexes on X and observe that at some point of Z, hence on a Zariski open subset, the complex is just the ideal sheaf of four points.

9 pages. Minor changes; to appear in Crelle as an appendix to 1305.0178

References in corpus (3)

• Homological projective duality for Grassmannians of lines
• Hochschild homology and semiorthogonal decompositions
• Lagrangian embeddings of cubic fourfolds containing a plane

Cited by in corpus (11)

• On the geometry of the LLSvS eightfold
• K3 categories, one-cycles on cubic fourfolds, and the Beauville-Voisin filtration
• On the period of Lehn, Lehn, Sorger, and van Straten's symplectic eightfold
• Automorphisms of positive entropy on some hyperKahler manifolds via derived automorphisms of K3 surfaces
• Lagrangian embeddings of cubic fourfolds containing a plane
• Twisted cubics and quadruples of points on cubic surfaces
• Non-symplectic automorphisms of odd prime order on manifolds of $K{3}^{[n]}$-type
• The indeterminacy locus of the Voisin map
• Hyper-Kaehler compactification of the intermediate Jacobian fibration of a cubic fourfold : the twisted case
• Kodaira dimension of universal holomorphic symplectic varieties
• A hyper-Kähler compactification of the Intermediate Jacobian fibration associated to a cubic fourfold