# Intermediate Jacobians and hyperKahler manifolds

## Giulia Sacca (Stonybrook)

In recent years, there have been an increasing number of connections between cubic 4folds and hyperkahler manifolds. The first instance of this was noticed by Beauville-Donagi, who showed that the Fano varieties of lines on a cubic 4folds X is holomorphic symplectic. The aim of the talk is to describe another instance of this phenomenon, which is carried out in joint work with R. Laza and C. Voisin. Given a general cubic 4fold X, we can consider the universal family Y_U \to U of smooth hyperplanes sections of X and the relative Intermediate Jacobian fibration f: J_U \to U. In 1995 Donagi and Markman constructed a holomorphic symplectic form on J_U, with respect to which the fibration f is Lagrangian. Since then, there have been many attempts to find a smooth hyperkahler compactification of J_U. This was conjectured to exist and to be deformation equivalent to O'Grady's 10--dimensional exceptional example. With Radu Laza and Claire Voisin, we solve this conjecture by using relative compactified Prym varieties.

Tuesday, October 25th, at 4:00pm in HBH 227