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⊂P5. 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 YH=Y∩H of Y give rise to a Lagrangian subvariety ZH⊂Z. For a generic choice of the hyperplane, ZH is birational to the theta-divisor in the intermediate Jacobian J(YH).