Fano manifolds of degree ten and EPW sextics

TL;DR

This paper links EPW sextics and Fano fourfolds of degree ten, showing how symplectic manifolds can be constructed from Hilbert schemes of conics, leading to new Lagrangian surfaces and integrable systems.

Contribution

It introduces a novel construction of symplectic manifolds from Fano fourfolds using Hilbert schemes of conics, providing new geometric insights.

Findings

Construction of symplectic manifolds from Fano fourfolds

Development of Lagrangian surfaces within these manifolds

Identification of related integrable systems with intermediate Jacobians

Abstract

O'Grady showed that certain special sextics in $\mathbb{P}^5$ called EPW sextics admit smooth double covers with a holomorphic symplectic structure. We propose another perspective on these symplectic manifolds, by showing that they can be constructed from the Hilbert schemes of conics on Fano fourfolds of degree ten. As applications, we construct families of Lagrangian surfaces in these symplectic fourfolds, and related integrable systems whose fibers are intermediate Jacobians.