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

For a smooth cubic fourfold Y , we study the moduli space M of semistable objects of Mukai vector 2λ1 + 2λ2 in the Kuznetsov component of Y . We show that with a certain choice of stability conditions, M admits a symplectic resolution ̃M , which is a smooth projective hyperk ̈ahler manifold, deformation equivalent to the 10-dimensional ex- amples constructed by O’Grady. As applications, we show that a birational model of ̃M provides a hyperk ̈ahler compactification of the twisted family of intermediate Jacobians as- sociated to Y . This generalizes the previous result of Voisin [Voi18] in the very general case. We also prove that ̃M is the MRC quotient of the main component of the Hilbert scheme of quintic elliptic curves in Y , confirming a conjecture of Castravet.