Comptes Rendus
no. 9-10
Algebraic Geometry
Stable bundles as Frobenius morphism direct image
[Faisceaux stables en tant quʼimages directes par le morphisme de Frobenius]

Présenté par : Claire Voisin

Congjun Liu1 ; Mingshuo Zhou1
1 Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, PR China
Comptes Rendus. Mathématique, Volume 351 (2013) no. 9-10, pp. 381-383.

Soient X une courbe projective lisse de genre g⩾2 définie sur un corps k algébriquement clos de caractéristique p>0, et F:X→X1 le morphisme de Frobenius relatif. On montre quʼun fibré vectoriel E sur X1 est lʼimage directe sous F dʼun certain fibré stable sur X si et seulement si lʼinstabilité de F⁎E est égale à (p−1)(2g−2).

Let X be a smooth projective curve of genus g⩾2 over an algebraically closed field k of characteristic p>0, and let F:X→X1 be the relative Frobenius morphism. We show that a vector bundle E on X1 is the direct image under F of some stable bundle on X if and only if the instability of F⁎E is equal to (p−1)(2g−2).

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2013.04.021
Congjun Liu 1 ; Mingshuo Zhou 1

1 Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, PR China 
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Congjun Liu; Mingshuo Zhou. Stable bundles as Frobenius morphism direct image. Comptes Rendus. Mathématique, Volume 351 (2013) no. 9-10, pp. 381-383. doi : 10.1016/j.crma.2013.04.021. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2013.04.021/

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[3] V. Mehta; C. Pauly Semistability of Frobenius direct images over curves, Bull. Soc. Math. Fr., Volume 135 (2007), pp. 105-117

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

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