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 Hprim4(Y) to Hprim2(Z) is a Hodge isometry. We describe the full H2(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 Hilb4(K3). We propose a conjecture on how to use Z to produce equivalences from A to the derived category of a K3 surface.