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MMP for Enriques pairs and singular Enriques varieties

Francesco Antonio Denisi, Ángel David Ríos Ortiz, Nikolaos Tsakanikas, Zhixin Xie

Abstract

We introduce and study the class of primitive Enriques varieties, whose smooth members are Enriques manifolds. We provide several examples and we demonstrate that this class is stable under the operations of the Minimal Model Program (MMP). In particular, given an Enriques manifold $Y$ and an effective $\mathbb{R}$-divisor $B_Y$ on $Y$ such that the pair $(Y,B_Y)$ is log canonical, we prove that any $(K_Y+B_Y)$-MMP terminates with a minimal model $(Y',B_{Y'})$ of $(Y,B_Y)$, where $Y'$ is a $\mathbb{Q}$-factorial primitive Enriques variety with canonical singularities. Finally, we investigate the asymptotic theory of Enriques manifolds.

MMP for Enriques pairs and singular Enriques varieties

Abstract

We introduce and study the class of primitive Enriques varieties, whose smooth members are Enriques manifolds. We provide several examples and we demonstrate that this class is stable under the operations of the Minimal Model Program (MMP). In particular, given an Enriques manifold and an effective -divisor on such that the pair is log canonical, we prove that any -MMP terminates with a minimal model of , where is a -factorial primitive Enriques variety with canonical singularities. Finally, we investigate the asymptotic theory of Enriques manifolds.
Paper Structure (25 sections, 37 theorems, 122 equations)

This paper contains 25 sections, 37 theorems, 122 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Reflexive differential forms
  4. Galois and quasi-étale covers
  5. Augmented irregularity
  6. Reminders about an MMP step
  7. Equivariant MMP and Galois quasi-étale covers
  8. Symplectic varieties
  9. Definitions and basic properties
  10. Symplectic varieties
  11. Irreducible symplectic varieties
  12. Primitive symplectic varieties
  13. Finite group actions on primitive symplectic varieties
  14. MMP for symplectic pairs
  15. Enriques varieties
  16. ...and 10 more sections

Key Result

Theorem 1.1

Let $X$ be a projective IHS manifold and let $B$ be an effective $\mathbb{R}$-divisor on $X$ such that $(X,B)$ is a log canonical pair. Then any $(K_X + B)$-MMP terminates with a minimal model $(X',B')$ of $(X,B)$, where $X'$ is a $\mathbb{Q}$-factorial primitive symplectic variety and $B'$ is a nef

Theorems & Definitions (99)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Lemma 2.1
  • Remark 2.2
  • Remark 2.3
  • Proposition 2.4: Hodge duality for klt spaces
  • Definition 2.5
  • Remark 2.6
  • ...and 89 more