Orbifold Chern classes inequalities and applications

Rousseau, Erwan; Taji, Behrouz

doi:10.5802/aif.3571

Orbifold Chern classes inequalities and applications
[Inégalités de classes de Chern orbifoldes et applications]

Rousseau, Erwan ¹ ; Taji, Behrouz ²

¹ Université de Brest CNRS UMR 6205 Laboratoire de Mathématiques de Bretagne Atlantique F-29200 Brest (France)
² School of Mathematics and Statistics – Red Centre The University of New South Wales NSW 2052 (Australia)

Annales de l'Institut Fourier, Tome 73 (2023) no. 6, pp. 2371-2410.

Résumé
Abstract

Dans cet article nous prouvons que pour une paire $(X, D)$ avec $X$ une variété de dimension $3$ et $D$ un diviseur de bord avec peu de singularités, si $(K_{X} + D)$ est mobile, alors la seconde classe de Chern orbifolde $c_{2}$ de $(X, D)$ est pseudoeffective. Cela généralise le résultat classique de Miyaoka sur la pseudoeffectivité de $c_{2}$ pour les modèles minimaux. Comme application, nous donnons une solution simple à la conjecture de non-annulation effective de Kawamata en dimension $3$ , où nous prouvons que $H^{0} (X, K_{X} + H) \neq 0$ , lorsque $K_{X} + H$ est nef et $H$ un diviseur de Cartier réduit, ample et effectif. De plus, nous étudions la conjecture de Lang–Vojta pour les sous-variétés de codimension $1$ et montrons que les variétés minimales de dimension $3$ de type général ont un nombre fini de sous-variétés de codimension $1$ Fano, Calabi–Yau ou abéliennes avec peu de singularités et dont les classes numériques appartiennent au cône mobile.

In this paper we prove that given a pair $(X, D)$ of a threefold $X$ and a boundary divisor $D$ with mild singularities, if $(K_{X} + D)$ is movable, then the orbifold second Chern class $c_{2}$ of $(X, D)$ is pseudoeffective. This generalizes the classical result of Miyaoka on the pseudoeffectivity of $c_{2}$ for minimal models. As an application, we give a simple solution to Kawamata’s effective non-vanishing conjecture in dimension $3$ , where we prove that $H^{0} (X, K_{X} + H) \neq 0$ , whenever $K_{X} + H$ is nef and $H$ is an ample, effective, reduced Cartier divisor. Furthermore, we study Lang–Vojta’s conjecture for codimension one subvarieties and prove that minimal threefolds of general type have only finitely many Fano, Calabi–Yau or Abelian subvarieties of codimension one that are mildly singular and whose numerical classes belong to the movable cone.

Reçu le : 2019-10-01
Révisé le : 2021-11-23
Accepté le : 2022-08-01
Publié le : 2023-10-26

DOI : 10.5802/aif.3571

Classification : 14E30, 14J70, 14B05
Keywords: Classification theory, Miyaoka–Yau inequality, Movable cone of divisors, Minimal Models, Effective non-vanishing, Lang–Vojta’s conjecture.
Mot clés : Théorie de la classification, Inégalité de Miyaoka–Yau, cône mobile de diviseurs, modèles minimaux, non-annulation effective, conjecture de Lang–Vojta.

Affiliations des auteurs :

Rousseau, Erwan ¹ ; Taji, Behrouz ²

Licence :

CC-BY-ND 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     title = {Orbifold {Chern} classes inequalities and applications},
     journal = {Annales de l'Institut Fourier},
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Rousseau, Erwan; Taji, Behrouz. Orbifold Chern classes inequalities and applications. Annales de l'Institut Fourier, Tome 73 (2023) no. 6, pp. 2371-2410. doi : 10.5802/aif.3571. https://aif.centre-mersenne.org/articles/10.5802/aif.3571/

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