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

Presented by: 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.

Abstract
Résumé

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.

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.

Received: 2014-03-22
Accepted: 2014-04-25
Published online: 2014-06-11

DOI: 10.1016/j.crma.2014.04.011

Author's affiliations:

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/

References
Cited by

[1] I. Biswas Tangent bundle of a complete intersection, Trans. Amer. Math. Soc., Volume 362 (2010) no. 6, pp. 3149-3160

[2] A. Grothendieck Cohomologie locale des faisceaux cohérents et théorèmes de Lefschetz locaux et globaux (SGA 2), Adv. Stud. Pure Math., vol. 2, North-Holland Publishing Co./Masson & Cie, Amsterdam/Paris, 1968

[3] K. Joshi Kodaira–Akizuki–Nakano vanishing: a variant, Bull. Lond. Math. Soc., Volume 32 (2000) no. 2, pp. 171-176

[4] G. Li; F. Yu Instability of truncated symmetric powers of sheaves, J. Algebra, Volume 386 (2013), pp. 176-189

[5] A. Noma Stability of Frobenius pull-backs of tangent bundles and generic injectivity of Gauss maps in positive characteristic, Compos. Math., Volume 106 (1997), pp. 61-70

[6] A. Noma Stability of Frobenius pull-backs of tangent bundles of weighted complete intersections, Math. Nachr., Volume 221 (2001), pp. 87-93

[7] T. Peternell; J. Wiśniewski On the stability of tangent bundles of Fano manifolds, J. Algebr. Geom., Volume 4 (1995), pp. 363-384

[8] X. Sun Direct images of bundles under Frobenius morphisms, Invent. Math., Volume 173 (2008) no. 2, pp. 427-447

[9] X. Sun Stability of sheaves of locally closed and exact forms, J. Algebra, Volume 324 (2010), pp. 1471-1482

Cited by Sources:

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

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