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 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.

On the period of Li, Pertusi, and Zhao’s symplectic variety

“…The same question for Fano varieties of lines of cubic fourfolds and for the so-called Lehn-Lehn-Sorger-van Straten symplectic eightfolds has been previously answered, by similar methods, in [Add16,Huy17] and [AG20,LPZ18]. From this point of view, the present work can be thought as a natural continuation of the aforementioned works.…”