On the geometry of the Lehn–Lehn–Sorger–van Straten eightfold

Abstract

In this article we make a few remarks about the geometry of the holomorphic symplectic manifold $Z$ constructed by Lehn, Lehn, Sorger, and van Straten as a two-step contraction of the variety of twisted cubic curves on a cubic fourfold $Y\subset {\mathbb{P}}^5$. We show that $Z$ is birational to a component of the moduli space of stable sheaves in the Calabi–Yau subcategory of the derived category of $Y$. Using this description we deduce that the twisted cubics contained in a hyperplane section $Y_H=Y\cap H$ of $Y$ give rise to a Lagrangian subvariety $Z_H\subset Z$. For a generic choice of the hyperplane, $Z_H$ is birational to the theta-divisor in the intermediate Jacobian $\mathrm{J}\left(Y_H\right)$.