MMP for Enriques pairs and singular Enriques varieties

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

doi:10.5802/jep.334

MMP for Enriques pairs and singular Enriques varieties
[MMP pour les paires d’Enriques et variétés d’Enriques singulières]

Francesco Antonio Denisi ¹ ; Ángel David Ríos Ortiz ² ; Nikolaos Tsakanikas ³ ; Zhixin Xie ⁴

¹Fachrichtung Mathematik, Campus, Gebäude E2.4, Universität des Saarlandes, 66123 Saarbrücken, Germany
²Université Paris-Saclay, CNRS, Laboratoire de Mathématiques d’Orsay, Bât. 307, 91405 Orsay, France
³Institut de Mathématiques (CAG), École Polytechnique Fédérale de Lausanne (EPFL), 1015 Lausanne, Switzerland
⁴Institut Élie Cartan de Lorraine, Université de Lorraine, 54506 Nancy, France

Journal de l’École polytechnique — Mathématiques, Tome 13 (2026), pp. 629-686

Abstract (VO)
Résumé

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 preserved 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 $(Y^{\prime },B_{Y^{\prime }})$ of $(Y,B_Y)$, where $Y^{\prime }$ is a $\mathbb{Q}$-factorial primitive Enriques variety with canonical singularities. Finally, we investigate the asymptotic theory of Enriques manifolds.

Nous introduisons et étudions la classe des variétés d’Enriques primitives, dont les membres lisses sont les variétés d’Enriques. Nous construisons des exemples et nous démontrons que cette classe est préservée sous les opérations du programme des modèles minimaux (MMP). En particulier, étant donné une variété d’Enriques $Y$ et un $\mathbb{R}$-diviseur effectif sur $Y$ tel que la paire $(Y,B_Y)$ est log canonique, nous montrons que tout $(K_Y+B_Y)$-MMP se termine avec un modèle minimal $(Y^{\prime },B_{Y^{\prime }})$ de $(Y,B_Y)$, où $Y^{\prime }$ est une variété d’Enriques primitive $\mathbb{Q}$-factorielle à singularités canoniques. Finalement, nous nous intéressons à la théorie asymptotique des diviseurs sur les variétés d’Enriques.

Reçu le : 2024-11-06
Accepté le : 2026-02-24
Publié le : 2026-03-13

DOI : 10.5802/jep.334

Classification : 14E30, 14J42, 14J28, 14L30
Keywords: Enriques manifolds, irreducible holomorphic symplectic manifolds, singular Enriques varieties, symplectic varieties, Minimal Model Program, termination of flips
Mots-clés : Variétés d’Enriques, variétés symplectiques holomorphes irréductibles, variétés d’Enriques singulières, variétés symplectiques, programme des modèles minimaux, terminaison des flips

Affiliations des auteurs :

Francesco Antonio Denisi ¹ ; Ángel David Ríos Ortiz ² ; Nikolaos Tsakanikas ³ ; Zhixin Xie ⁴

¹ Fachrichtung Mathematik, Campus, Gebäude E2.4, Universität des Saarlandes, 66123 Saarbrücken, Germany
² Université Paris-Saclay, CNRS, Laboratoire de Mathématiques d’Orsay, Bât. 307, 91405 Orsay, France
³ Institut de Mathématiques (CAG), École Polytechnique Fédérale de Lausanne (EPFL), 1015 Lausanne, Switzerland
⁴ Institut Élie Cartan de Lorraine, Université de Lorraine, 54506 Nancy, France

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

@article{JEP_2026__13__629_0,
     author = {Francesco Antonio Denisi and \'Angel David R{\'\i}os Ortiz and Nikolaos Tsakanikas and Zhixin Xie},
     title = {MMP for {Enriques} pairs and {singular~Enriques~varieties}},
     journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
     pages = {629--686},
     year = {2026},
     publisher = {\'Ecole polytechnique},
     volume = {13},
     doi = {10.5802/jep.334},
     language = {en},
     url = {https://jep.centre-mersenne.org/articles/10.5802/jep.334/}
}

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Francesco Antonio Denisi; Ángel David Ríos Ortiz; Nikolaos Tsakanikas; Zhixin Xie. MMP for Enriques pairs and singular Enriques varieties. Journal de l’École polytechnique — Mathématiques, Tome 13 (2026), pp. 629-686. doi: 10.5802/jep.334

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