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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
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 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 OG10–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) H2,2(X, )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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