On the geometry of the LLSvS eightfold

TL;DR

This paper explores the geometric structure of a special holomorphic symplectic manifold related to cubic fourfolds, revealing its birational relations to moduli spaces and Lagrangian subvarieties connected to hyperplane sections.

Contribution

It establishes a birational equivalence between the symplectic manifold Z and a moduli space of stable sheaves, and describes the Lagrangian subvarieties arising from hyperplane sections.

Findings

Z is birational to a moduli space of stable sheaves.

Hyperplane sections induce Lagrangian subvarieties in Z.

For generic hyperplanes, Z_H is birational to a theta-divisor.

Abstract

In this note we make a few remarks about the geometry of the holomorphic symplectic manifold Z constructed by C.Lehn, M.Lehn, C.Sorger and D. van Straten as a two-step contraction of the variety of twisted cubic curves on a cubic fourfold Y in P^5. We show that Z is birational to a component of a 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 of Y give rise to a Lagrangian subvariety Z_H in Z. For a generic choice of the hyperplane, Z_H is birational to the theta-divisor in the intermediate Jacobian of Y_H.