The stability of Frobenius direct images of rank-two bundles over surfaces

Liu, Congjun; Zhou, Mingshuo

doi:10.1016/j.crma.2014.12.001

Algebraic geometry

The stability of Frobenius direct images of rank-two bundles over surfaces
[La stabilité de l'image directe de Frobenius des fibrés de rang deux sur des surfaces]

Présenté par : Claire Voisin

Liu, Congjun ¹ ; Zhou, Mingshuo ²

¹ Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, PR China
² School of Science, Hangzhou Dianzi University, Hangzhou 310018, PR China

Comptes Rendus. Mathématique, Tome 353 (2015) no. 4, pp. 339-344.

Résumé
Abstract

Soit X une surface projective lisse sur un corps algébriquement clos k de caractéristique $p \geq 5$ avec $Ω_{X}^{1}$ semistable et $μ (Ω_{X}^{1}) > 0$ . Étant donné un fibré vectoriel semistable (resp. stable) W de rang 2 sur X, on montre que l'image directe $F_{⁎} W$ par le morphisme de Frobenius F est aussi semistable (resp. stable).

Let X be a smooth projective surface over an algebraically closed field k of characteristic $p \geq 5$ with $Ω_{X}^{1}$ semistable and $μ (Ω_{X}^{1}) > 0$ . Given a semistable (resp. stable) vector bundle W of rank 2, we prove that the direct image $F_{⁎} W$ under the Frobenius morphism F is also semistable (resp. stable).

Reçu le : 2014-04-28
Accepté le : 2014-12-11
Publié le : 2015-02-18

DOI : 10.1016/j.crma.2014.12.001

Affiliations des auteurs :

Liu, Congjun ¹ ; Zhou, Mingshuo ²

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     title = {The stability of {Frobenius} direct images of rank-two bundles over surfaces},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {339--344},
     publisher = {Elsevier},
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Liu, Congjun; Zhou, Mingshuo. The stability of Frobenius direct images of rank-two bundles over surfaces. Comptes Rendus. Mathématique, Tome 353 (2015) no. 4, pp. 339-344. doi : 10.1016/j.crma.2014.12.001. http://www.numdam.org/articles/10.1016/j.crma.2014.12.001/

Bibliographie
Cité par

[1] Huybrechts, D.; Lehn, M. The Geometry of Moduli Spaces of Sheaves, Aspects Math., vol. 31, Friedr, Vieweg Sohn, Braunschweig, 1997

[2] Joshi, K.; Ramanan, S.; Xia, E.; Yu, J.-K. On vector bundles destabilized by Frobenius pull-back, Compositio Math., Volume 142 (2006) no. 3, pp. 616-630

[3] Kitadai, Y.; Sumihiro, H. Canonical filtrations and stability of direct images by Frobenius morphism II, Hiroshima Math. J., Volume 38 (2008), pp. 243-261

[4] Langer, A. Semistable sheaves in positive characteristic, Ann. of Math. (2), Volume 159 (2004), pp. 251-276

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

[6] Sun, X. Direct images of bundles under Frobenius morphism, Invent. Math., Volume 173 (2008), pp. 427-447

[7] Sun, X. Frobenius morphism and semistable bundles, Algebraic Geometry in East Asia (2008), pp. 161-182

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