Kyoto Journal of Mathematics

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

Evgeny Shinder and Andrey Soldatenkov

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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 YP5. 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=YH of Y give rise to a Lagrangian subvariety ZHZ. For a generic choice of the hyperplane, ZH is birational to the theta-divisor in the intermediate Jacobian J(YH).

Article information

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

Received: 5 April 2016
Accepted: 10 June 2016
First available in Project Euclid: 9 June 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14F05: Sheaves, derived categories of sheaves and related constructions [See also 14H60, 14J60, 18F20, 32Lxx, 46M20]
Secondary: 53C26: Hyper-Kähler and quaternionic Kähler geometry, "special" geometry

moduli spaces of sheaves irreducible holomorphic symplectic manifolds cubic fourfolds Atiyah class


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.

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