Download this article
 Download this article For screen
For printing
Recent Issues

Volume 28
Issue 3, 1005–1499
Issue 2, 497–1003
Issue 1, 1–496

Volume 27, 9 issues

Volume 26, 8 issues

Volume 25, 7 issues

Volume 24, 7 issues

Volume 23, 7 issues

Volume 22, 7 issues

Volume 21, 6 issues

Volume 20, 6 issues

Volume 19, 6 issues

Volume 18, 5 issues

Volume 17, 5 issues

Volume 16, 4 issues

Volume 15, 4 issues

Volume 14, 5 issues

Volume 13, 5 issues

Volume 12, 5 issues

Volume 11, 4 issues

Volume 10, 4 issues

Volume 9, 4 issues

Volume 8, 3 issues

Volume 7, 2 issues

Volume 6, 2 issues

Volume 5, 2 issues

Volume 4, 1 issue

Volume 3, 1 issue

Volume 2, 1 issue

Volume 1, 1 issue

The Journal
About the Journal
Editorial Board
Editorial Procedure
Submission Guidelines
Submission Page
Policies for Authors
Ethics Statement
ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
Author Index
To Appear
Other MSP Journals
Birational geometry of the intermediate Jacobian fibration of a cubic fourfold

Giulia Saccà

Appendix: Claire Voisin

Geometry & Topology 27 (2023) 1479–1538

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.

hyper-Kähler, holomorphic symplectic, OG10, intermediate Jacobian
Mathematical Subject Classification
Primary: 14D06, 14J42
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
Giulia Saccà
Department of Mathematics
Columbia University
New York, NY
United States
Claire Voisin
Institut de mathématiques de Jussieu

Open Access made possible by participating institutions via Subscribe to Open.