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Gushel--Mukai varieties: intermediate Jacobians

Olivier Debarre, Alexander Kuznetsov

Abstract

We describe intermediate Jacobians of Gushel-Mukai varieties $X$ of dimensions 3 or 5: if $A$ is the Lagrangian space associated with $X$, we prove that the intermediate Jacobian of $X$ is isomorphic to the Albanese variety of the canonical double covering of any of the two dual Eisenbud-Popescu-Walter surfaces associated with $A$. As an application, we describe the period maps for Gushel-Mukai threefolds and fivefolds.

Gushel--Mukai varieties: intermediate Jacobians

Abstract

We describe intermediate Jacobians of Gushel-Mukai varieties of dimensions 3 or 5: if is the Lagrangian space associated with , we prove that the intermediate Jacobian of is isomorphic to the Albanese variety of the canonical double covering of any of the two dual Eisenbud-Popescu-Walter surfaces associated with . As an application, we describe the period maps for Gushel-Mukai threefolds and fivefolds.

Paper Structure

This paper contains 27 sections, 48 theorems, 240 equations.

Table of Contents

  1. Introduction
  2. Gushel--Mukai and Eisenbud--Popescu--Walter varieties
  3. Geometry of $\mathop{\mathrm{\mathsf{Gr}}}\nolimits(2,5)$
  4. Eisenbud--Popescu--Walter varieties and their double coverings
  5. Gushel--Mukai varieties
  6. Linear spaces on quadrics containing GM varieties
  7. Linear spaces on ordinary GM threefolds and fivefolds
  8. $\sigma$-planes on ordinary GM fivefolds
  9. Lines on ordinary GM threefolds
  10. Topological preliminaries
  11. Abel--Jacobi maps
  12. Generalized blow up decomposition
  13. The Clemens--Tyurin argument
  14. Intermediate Jacobians and their endomorphisms
  15. Intermediate Jacobians of GM threefolds
  16. ...and 12 more sections

Key Result

Theorem 1.1

For any Lagrangian subspace $A \subset {\bigwedge ^{3}} {V}_6$ with no decomposable vectors, the Albanese variety $\mathop{\mathrm{Alb}}\nolimits(\widetilde{Y}^{\ge2}_{A})$ has a canonical principal polarization such that there is an isomorphism of principally polarized abelian varieties. If $X$ is a smooth GM variety of dimension $n \in \{3,5\}$, with intermediate Jacobian $\mathop{\mathrm{Jac}}

Theorems & Definitions (98)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • Proposition 2.5
  • proof
  • ...and 88 more