Semistability of Frobenius direct images over curves

Mehta, Vikram B.; Pauly, Christian

doi:10.24033/bsmf.2528

Semistability of Frobenius direct images over curves
[Semi-stabilité des images directes par Frobenius sur les courbes]

Mehta, Vikram B. ; Pauly, Christian

Bulletin de la Société Mathématique de France, Tome 135 (2007) no. 1, pp. 105-117.

Résumé
Abstract

Soit $X$ une courbe projective lisse de genre $\geq 2$ définie sur un corps $k$ algébriquement clos de caractéristique $p > 0$ . Étant donné un fibré vectoriel semi-stable $E$ sur $X$ , nous montrons que l’image directe $F_{*} E$ par le morphisme de Frobenius $F$ de $X$ est aussi semi-stable. Nous déduisons une caractérisation numérique du fibré vectoriel stable $F_{*} L$ de rang $p$ , où $L$ est un fibré en droites sur $X$ .

Let $X$ be a smooth projective curve of genus $g \geq 2$ defined over an algebraically closed field $k$ of characteristic $p > 0$ . Given a semistable vector bundle $E$ over $X$ , we show that its direct image $F_{*} E$ under the Frobenius map $F$ of $X$ is again semistable. We deduce a numerical characterization of the stable rank- $p$ vector bundles $F_{*} L$ , where $L$ is a line bundle over $X$ .

EuDML MR Zbl

DOI : 10.24033/bsmf.2528

Classification : 14H40, 14D20, 14H40
Keywords: vector bundle, semistability, Frobenius
Mot clés : fibré vectoriel, semi-stabilité, Frobenius

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     title = {Semistability of {Frobenius} direct images over curves},
     journal = {Bulletin de la Soci\'et\'e Math\'ematique de France},
     pages = {105--117},
     publisher = {Soci\'et\'e math\'ematique de France},
     volume = {135},
     number = {1},
     year = {2007},
     doi = {10.24033/bsmf.2528},
     mrnumber = {2430201},
     zbl = {1201.14021},
     language = {en},
     url = {https://www.numdam.org/articles/10.24033/bsmf.2528/}
}

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Mehta, Vikram B.; Pauly, Christian. Semistability of Frobenius direct images over curves. Bulletin de la Société Mathématique de France, Tome 135 (2007) no. 1, pp. 105-117. doi : 10.24033/bsmf.2528. https://www.numdam.org/articles/10.24033/bsmf.2528/

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