Elliptic quintics on cubic fourfolds, O'Grady 10, and Lagrangian fibrations

Abstract

For a smooth cubic fourfold Y, we study the moduli space M of semistable objects of Mukai vector $2{\lambda }_1+2{\lambda }_2$ in the Kuznetsov component of Y. We show that with a certain choice of stability conditions, M admits a symplectic resolution $\tilde{M}$, which is a smooth projective hyperkähler manifold, deformation equivalent to the 10-dimensional examples constructed by O'Grady. As applications, we show that a birational model of $\tilde{M}$ provides a hyperkähler compactification of the twisted family of intermediate Jacobians associated to Y. This generalizes the previous result of Voisin arXiv:1611.06679 in the very general case. We also prove that $\tilde{M}$ is the MRC quotient of the main component of the Hilbert scheme of elliptic quintic curves in Y, confirming a conjecture of Castravet.

42 pages