Birational geometry of the intermediate Jacobian fibration of a cubic fourfold

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 $J\to {\mathbb{P}}^5$. This builds upon arXiv:1602.05534, where the result is proved for general $X$, as well as on the degeneration techniques on arXiv:1704.02731 and techniques from 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 arXiv:1611.06679, 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 $MW(\pi )$ of the abelian fibration $\pi :J\to {\mathbb{P}}^5$ is isomorphic to the integral degree $4$ primitive algebraic cohomology of $X$, i.e., $MW(\pi )=H^{2,2}(X,\mathbb{Z}{)}_0$.

34 pages; minor expository improvements following the suggestion of the referee; accepted by Geom. Topology

References in corpus (1)

• Quelques remarques sur la décomposition de Zariski divisorielle sur les variétés dont la première classe de Chern est nulle

Cited by in corpus (4)

• On the monodromy group of desingularised moduli spaces of sheaves on K3 surfaces
• The dual Lagrangian fibration of known hyper-Kähler manifolds
• Relative and absolute Lefschetz standard conjectures for some Lagrangian fibrations
• The Intermediate Jacobian fibration of a cubic fourfold containing a plane and fibrations in Prym varieties