Strong stability of cotangent bundles of cyclic covers

Lingguang Li; Junchao Shentu

doi:10.1016/j.crma.2014.04.011

Algebraic geometry

Strong stability of cotangent bundles of cyclic covers
[Stabilité forte du fibré cotangent des revêtements cycliques]

Présenté par : Claire Voisin

Lingguang Li ¹ ; Junchao Shentu ²

¹ Department of Mathematics, Tongji University, Shanghai, PR China
² Academy of Mathematics and Systems Science, Chinese Academy of Science, Beijing, PR China

Comptes Rendus. Mathématique, Volume 352 (2014) no. 7-8, pp. 639-644.

Résumé
Abstract

Soit X une varieté projective lisse sur un corps algébriquement clos k de caractéristique $p > 0$ de dimension $\dim X \geq 4$ et avec nombre de Picard $ρ (X) = 1$ . Supposons que X satisfasse $H^{i} (X, F_{X}^{m ⁎} (Ω_{X}^{j}) \otimes L^{- 1}) = 0$ pour tout fibré en droite ample $L$ sur X et tous nombres entiers $m, i, j$ tels que $0 \leq i + j < \dim X$ , où $F_{X} : X \to X$ est le morphisme de Frobenius absolu. Soit Y une varieté lisse obtenue par X en prenant des sections hyperplanes lisses de dimension ≥3 et des revêtements cycliques le long des diviseurs lisses. Si le fibré canonique $ω_{Y}$ est ample (resp. nef), alors on montre que $Ω_{Y}$ est fortement stable (resp. fortement semistable) par rapport à n'importe quelle polarisation.

Let X be a smooth projective variety over an algebraically closed field k of characteristic $p > 0$ of $\dim X \geq 4$ and Picard number $ρ (X) = 1$ . Suppose that X satisfies $H^{i} (X, F_{X}^{m ⁎} (Ω_{X}^{j}) \otimes L^{- 1}) = 0$ for any ample line bundle $L$ on X, and any nonnegative integers $m, i, j$ with $0 \leq i + j < \dim X$ , where $F_{X} : X \to X$ is the absolute Frobenius morphism. Let Y be a smooth variety obtained from X by taking hyperplane sections of dim ≥3 and cyclic covers along smooth divisors. If the canonical bundle $ω_{Y}$ is ample (resp. nef), then we prove that $Ω_{Y}$ is strongly stable (resp. strongly semistable) with respect to any polarization.

Reçu le : 2014-03-22
Accepté le : 2014-04-25
Publié le : 2014-06-11

DOI : 10.1016/j.crma.2014.04.011

Affiliations des auteurs :

Lingguang Li ¹ ; Junchao Shentu ²

¹ Department of Mathematics, Tongji University, Shanghai, PR China
² Academy of Mathematics and Systems Science, Chinese Academy of Science, Beijing, PR China

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     title = {Strong stability of cotangent bundles of cyclic covers},
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Lingguang Li; Junchao Shentu. Strong stability of cotangent bundles of cyclic covers. Comptes Rendus. Mathématique, Volume 352 (2014) no. 7-8, pp. 639-644. doi : 10.1016/j.crma.2014.04.011. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2014.04.011/

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Cité par Sources :

^☆ The first author is supported by the National Natural Science Foundation of China (No. 11271275).

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