On the period of Lehn, Lehn, Sorger, and van Straten's symplectic eightfold

TL;DR

This paper investigates the Hodge-theoretic properties of a specific eight-dimensional symplectic manifold associated with a cubic fourfold, revealing isometries and conditions for birationality to moduli spaces of sheaves.

Contribution

It establishes a Hodge isometry via the Abel-Jacobi map, describes the second cohomology in terms of the Mukai lattice, and proposes a conjecture relating the symplectic eightfold to derived categories of K3 surfaces.

Findings

Hodge isometry between H^4_prim(Y) and H^2_prim(Z)

Full description of H^2(Z) via Mukai lattice

Numerical conditions for Z to be birational to moduli spaces of sheaves or Hilbert schemes

Abstract

For the irreducible holomorphic symplectic eightfold Z associated to a cubic fourfold Y not containing a plane, we show that a natural Abel-Jacobi map from H^4_prim(Y) to H^2_prim(Z) is a Hodge isometry. We describe the full H^2(Z) in terms of the Mukai lattice of the K3 category A of Y. We give numerical conditions for Z to be birational to a moduli space of sheaves on a K3 surface or to Hilb^4(K3). We propose a conjecture on how to use Z to produce equivalences from A to the derived category of a K3 surface.