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

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 $H_{\mathrm{prim}}^4(Y)$ to $H_{\mathrm{prim}}^2(Z)$ is a Hodge isometry. We describe the full $H^2(Z)$ in terms of the Mukai lattice of the K3 category $\mathcal{A}$ of Y. We give numerical conditions for Z to be birational to a moduli space of sheaves on a K3 surface or to ${\mathrm{Hilb}}^4$(K3). We propose a conjecture on how to use Z to produce equivalences from $\mathcal{A}$ to the derived category of a K3 surface.