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

TL;DR

This paper constructs a new hyperk"ahler manifold from a cubic fourfold's moduli space, extending known examples and connecting it to elliptic quintic curves and intermediate Jacobians.

Contribution

It introduces a symplectic resolution of a moduli space in the Kuznetsov component, linking it to O'Grady's hyperk"ahler examples and confirming a conjecture about elliptic quintic curves.

Findings

Construction of a hyperk"ahler manifold deformation equivalent to O'Grady's examples.

Establishment of a birational model as a hyperk"ahler compactification of twisted intermediate Jacobians.

Confirmation that the main component of the elliptic quintic Hilbert scheme's MRC quotient is this hyperk"ahler manifold.

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 hyperk\"ahler 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 hyperk\"ahler 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.