## Acta Mathematica

### A hyper-Kähler compactification of the intermediate Jacobian fibration associated with a cubic 4-fold

#### Abstract

Let $X$ be a general cubic $4$-fold. It was observed by Donagi and Markman that the relative intermediate Jacobian fibration $\mathcal{J}_U/U$ (with $U=(\mathbb{P}^5)^\vee\setminus X^\vee$) associated with the family of smooth hyperplane sections of $X$ carries a natural holomorphic symplectic form making the fibration Lagrangian. In this paper, we obtain a smooth projective compactification $\overline{\mathcal{J}}$ of $\mathcal{J}_U$ with the property that the holomorphic symplectic form on $\mathcal{J}_U$ extends to a holomorphic symplectic form on $\overline{\mathcal{J}}$. In particular, $\overline{\mathcal{J}}$ is a $10$-dimensional compact hyper-Kähler manifold, which we show to be deformation equivalent to the exceptional example of O'Grady. This proves a conjecture by Kuznetsov and Markushevich.

#### Article information

Source
Acta Math., Volume 218, Number 1 (2017), 55-135.

Dates
Revised: 31 October 2016
First available in Project Euclid: 14 September 2017

https://projecteuclid.org/euclid.acta/1505401887

Digital Object Identifier
doi:10.4310/ACTA.2017.v218.n1.a2

Mathematical Reviews number (MathSciNet)
MR3710794

Zentralblatt MATH identifier
06826204

#### Citation

Laza, Radu; Saccà, Giulia; Voisin, Claire. A hyper-Kähler compactification of the intermediate Jacobian fibration associated with a cubic 4-fold. Acta Math. 218 (2017), no. 1, 55--135. doi:10.4310/ACTA.2017.v218.n1.a2. https://projecteuclid.org/euclid.acta/1505401887