## Kyoto Journal of Mathematics

### 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 $\mathrm{J}(Y_{H})$.

#### Article information

Source
Kyoto J. Math., Volume 57, Number 4 (2017), 789-806.

Dates
Accepted: 10 June 2016
First available in Project Euclid: 9 June 2017

https://projecteuclid.org/euclid.kjm/1496973624

Digital Object Identifier
doi:10.1215/21562261-2017-0014

Mathematical Reviews number (MathSciNet)
MR3725260

Zentralblatt MATH identifier
06825577

#### Citation

Shinder, Evgeny; Soldatenkov, Andrey. On the geometry of the Lehn–Lehn–Sorger–van Straten eightfold. Kyoto J. Math. 57 (2017), no. 4, 789--806. doi:10.1215/21562261-2017-0014. https://projecteuclid.org/euclid.kjm/1496973624

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