Comptes Rendus
no. 4
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

Congjun Liu1 ; Mingshuo Zhou2
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, Volume 353 (2015) no. 4, pp. 339-344.

Soit X une surface projective lisse sur un corps algébriquement clos k de caractéristique p≥5 avec ΩX1 semistable et μ(ΩX1)>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≥5 with ΩX1 semistable and μ(ΩX1)>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 :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2014.12.001
Congjun Liu 1 ; Mingshuo Zhou 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 
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Congjun Liu; Mingshuo Zhou. The stability of Frobenius direct images of rank-two bundles over surfaces. Comptes Rendus. Mathématique, Volume 353 (2015) no. 4, pp. 339-344. doi : 10.1016/j.crma.2014.12.001. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2014.12.001/

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

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

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

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

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

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

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

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