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Logarithmic Enriques varieties

Samuel Boissiere, Chiara Camere, Alessandra Sarti

TL;DR

This work defines log-Enriques varieties as singular analogues of Enriques manifolds, tied to K-torsion canonical divisors and zero augmented irregularity, and develops a framework in which their canonical covers canonically decompose into ISV/PSV/CY factors. It then focuses on log-Enriques varieties of symplectic type, showing that cyclic quasi-étale quotients by purely nonsymplectic automorphisms yield rich families of examples with computable indices and geometric invariants. The paper provides an extensive catalog of constructions: Enriques manifolds and log-Enriques varieties arising from Hilbert schemes, generalized Kummer varieties, moduli spaces of sheaves on K3, complete intersections, and Prym-type objects, across IHS, ISV, PSV, and weak CY settings. These results broaden the landscape of higher-dimensional Enriques-type geometries and offer systematic pathways to producing and studying singular analogues with controlled canonical classes and singularities, with potential impact on moduli theory and birational geometry of Chern-class-zero varieties.

Abstract

We introduce logarithmic Enriques varieties as a singular analogue of Enriques manifolds, generalizing the notion of log-Enriques surfaces introduced by Zhang. We focus then on the properties of the subfamily of log-Enriques varieties that admit a quasi-étale cover by a singular symplectic variety and we give many examples.

Logarithmic Enriques varieties

TL;DR

This work defines log-Enriques varieties as singular analogues of Enriques manifolds, tied to K-torsion canonical divisors and zero augmented irregularity, and develops a framework in which their canonical covers canonically decompose into ISV/PSV/CY factors. It then focuses on log-Enriques varieties of symplectic type, showing that cyclic quasi-étale quotients by purely nonsymplectic automorphisms yield rich families of examples with computable indices and geometric invariants. The paper provides an extensive catalog of constructions: Enriques manifolds and log-Enriques varieties arising from Hilbert schemes, generalized Kummer varieties, moduli spaces of sheaves on K3, complete intersections, and Prym-type objects, across IHS, ISV, PSV, and weak CY settings. These results broaden the landscape of higher-dimensional Enriques-type geometries and offer systematic pathways to producing and studying singular analogues with controlled canonical classes and singularities, with potential impact on moduli theory and birational geometry of Chern-class-zero varieties.

Abstract

We introduce logarithmic Enriques varieties as a singular analogue of Enriques manifolds, generalizing the notion of log-Enriques surfaces introduced by Zhang. We focus then on the properties of the subfamily of log-Enriques varieties that admit a quasi-étale cover by a singular symplectic variety and we give many examples.
Paper Structure (22 sections, 13 theorems, 25 equations)

This paper contains 22 sections, 13 theorems, 25 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. K-torsion varieties
  4. Enriques manifolds and log-Enriques varieties
  5. Log-Enriques varieties of symplectic type
  6. Examples
  7. Known examples of Enriques manifolds
  8. The Hilbert scheme of points on Enriques surfaces.
  9. Involutions on generalized Kummer varieties of decomposable abelian surfaces.
  10. Enriques involutions on K3 surfaces
  11. Index $2$ Enriques manifolds from moduli spaces of stable sheaves on $K3$ surfaces.
  12. Complete intersections of quadrics
  13. Index $3$ Enriques manifolds from generalized Kummer varieties of special decomposable abelian surfaces.
  14. Index $4$ Enriques manifolds from generalized Kummer manifolds of special decomposable abelian surfaces.
  15. Log-Enriques varieties of IHS type
  16. ...and 7 more sections

Key Result

Proposition 2.1

Let $X$ be a normal projective variety with rational singularities and such that $\omega_X\cong \mathcal{O}_X$, and assume that $\pi_1(X_{{\rm reg}})=\{1\}$.

Theorems & Definitions (36)

  • Proposition 2.1
  • proof
  • Definition 3.1
  • Proposition 3.2
  • proof
  • Definition 4.1
  • Definition 4.2
  • Lemma 4.3
  • proof
  • Lemma 4.4
  • ...and 26 more