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EPW sextics vs EPW cubes

Grzegorz Kapustka, Michal Kapustka, Giovanni Mongardi

Abstract

We study a correspondence between double EPW cubes and double EPW sextics, two families of polarized hyper-Kähler 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.

EPW sextics vs EPW cubes

Abstract

We study a correspondence between double EPW cubes and double EPW sextics, two families of polarized hyper-Kähler 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.
Paper Structure (10 sections, 21 theorems, 78 equations)

This paper contains 10 sections, 21 theorems, 78 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The incidence correspondence
  4. Elliptic curves of minimal degree
  5. Hodge structures
  6. Period domains and period maps
  7. Period maps
  8. The image of the period maps
  9. The case $\mathbb{S}_4$
  10. The cases $\mathbb{S}_2^{\ast}$ and $\mathbb{S}^{\#}_2$.

Key Result

Theorem 1.1

Let $A$ be a general Lagrangian subspace of $\bigwedge^3 W$ and let $\widetilde{X}_A$ and $\widetilde{Y}_A$ be the double EPW sextic and cube associated to $A$. Then the polarized integral Hodge structures $H^2_{{\operatorname{prim}}}(\widetilde{X}_A,\mathbb{Z})$ and $H^2_{{\operatorname{prim}}}(\wi

Theorems & Definitions (48)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Remark 3.3
  • Proposition 3.4
  • proof
  • ...and 38 more