EPW sextics vs EPW cubes

TL;DR

This paper explores the relationship between double EPW cubes and double EPW sextics, revealing their connections through Hodge structures and moduli spaces, and demonstrates that a general double EPW cube is a moduli space of stable objects.

Contribution

It establishes a correspondence between double EPW cubes and sextics and proves that a general double EPW cube is a moduli space of stable objects in a derived category.

Findings

Established a relation between double EPW cubes and sextics via Hodge structures.

Proved that a very general double EPW cube is a moduli space of stable objects.

Connected the geometry of EPW varieties to Gushel--Mukai fourfolds.

Abstract

We study a correspondence between double EPW cubes and double EPW sextics, two families of polarized hyper-K\"ahler manifolds related to Gushel--Mukai fourfolds. We infer relations between these families in terms of Hodge structures and moduli spaces of elliptic curves. As an application, we prove that a very general double EPW cube is the moduli space of stable objects with respect to a suitable stability condition on the Kuznetsov component of its corresponding Gushel--Mukai fourfolds; this answers a problem posed by Perry, Pertusi and Zhao.