Birational geometry of the intermediate Jacobian fibration of a cubic fourfold

Saccà, Giulia

doi:10.2140/gt.2023.27.1479

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Birational geometry of the intermediate Jacobian fibration of a cubic fourfold

Giulia Saccà

Appendix: Claire Voisin

Geometry & Topology 27 (2023) 1479–1538

DOI: 10.2140/gt.2023.27.1479

Abstract

We show that the intermediate Jacobian fibration associated to any smooth cubic fourfold $X$ admits a hyper-Kähler compactification $J (X)$ with a regular Lagrangian fibration $π : J \to ℙ^{5}$ . This builds upon work of Laza, Saccà and Voisin (2017), where the result is proved for general $X$ , as well as on the degeneration techniques introduced in the work of Kollár, Laza, Saccà and Voisin, and the minimal model program. We then study some aspects of the birational geometry of $J (X)$ : for very general $X$ we compute the movable and nef cones of $J (X)$ , showing that $J (X)$ is not birational to the twisted version of the intermediate Jacobian fibration, nor to an OG $10$ –type moduli space of objects in the Kuznetsov component of $X$ ; for any smooth $X$ we show, using normal functions, that the Mordell–Weil group $MW (π)$ of the fibration is isomorphic to the integral degree- $4$ primitive algebraic cohomology of $X$ , ie $MW (π) ≅ H^{2, 2} {(X, ℤ)}_{0}$ .

Keywords

hyper-Kähler, holomorphic symplectic, OG10, intermediate Jacobian

Mathematical Subject Classification

Primary: 14D06, 14J42

References

Publication

Received: 5 January 2021

Revised: 8 November 2021

Accepted: 8 December 2021

Published: 15 June 2023

Proposed: Richard P Thomas

Seconded: Mark Gross, Dan Abramovich

Authors

Giulia Saccà
Department of Mathematics Columbia University New York, NY United States

Claire Voisin
Institut de mathématiques de Jussieu Paris France

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Volume 27, issue 4 (2023)