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

Congjun Liu; Mingshuo Zhou

doi:10.1016/j.crma.2014.12.001

Algebraic geometry

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

Presented by: Claire Voisin

Congjun Liu ¹; Mingshuo Zhou ²

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

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

Abstract (Original language)
Résumé

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).

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).

Received: 2014-04-28
Accepted: 2014-12-11
Published online: 2015-02-18

DOI: 10.1016/j.crma.2014.12.001

Author's affiliations:

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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     author = {Congjun Liu and Mingshuo Zhou},
     title = {The stability of {Frobenius} direct images of rank-two bundles over surfaces},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {339--344},
     year = {2015},
     publisher = {Elsevier},
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     number = {4},
     doi = {10.1016/j.crma.2014.12.001},
     language = {en},
}

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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

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