Comptes Rendus
no. 9-10
Algebraic Geometry
Stable bundles as Frobenius morphism direct image
Presented by: Claire Voisin
Congjun Liu1; Mingshuo Zhou1
1Institute 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

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).

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).

Received:
Accepted:
Published online:
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

[1] D. Gieseker Stable vector bundles and the Frobenius morphism, Ann. Sci. Éc. Norm. Super. (4), Volume 6 (1973), pp. 95-101

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

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

[4] X. Sun Remarks on semistability of G-bundles in positive characteristic, Compos. Math., Volume 119 (1999), pp. 41-52

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

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