The construction of compact hyperkahler (HK) manifolds is a notoriously difficult problem. Currently, all the known examples are: two infinite series (deformations of Hilbert schemes of points on K3s and generalized Kummer varieties) and two exotic examples due to O'Grady in dimension 6 (OG6) and dimension 10 (OG10). While O'Grady's construction of OG10 is based on Mukai's approach via moduli of sheaves on K3s, we propose here a new "Lagrangian" construction for OG10. Specifically, we start with a general cubic fourfold X, and consider the intermediate Jacobian fibration J associated to the universal family of hyperplane sections on X. This is well-defined and algebraic (cf. Donagi-Markman) over the locus U of smooth hyperplane sections of X. As previously conjectured by Markushevich, we prove that J/U admits a smooth, flat compactification, which is a hyperkahler manifold, deformation equivalent to OG10. Our main tool here is the construction of a relative compactified Prym.
This is joint work with G. Sacca and C. Voisin.