Semistability of Frobenius direct images over curves

Vikram B. Mehta; Christian Pauly

Bulletin de la Société Mathématique de France (2007)

  • Volume: 135, Issue: 1, page 105-117
  • ISSN: 0037-9484

Abstract

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Let X be a smooth projective curve of genus g ≥ 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 .

How to cite

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Mehta, Vikram B., and Pauly, Christian. "Semistability of Frobenius direct images over curves." Bulletin de la Société Mathématique de France 135.1 (2007): 105-117. <http://eudml.org/doc/272383>.

@article{Mehta2007,
abstract = {Let $X$ be a smooth projective curve of genus $g \ge 2$ defined over an algebraically closed field $k$ of characteristic $p&gt;0$. 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 $X$.},
author = {Mehta, Vikram B., Pauly, Christian},
journal = {Bulletin de la Société Mathématique de France},
keywords = {vector bundle; semistability; Frobenius},
language = {eng},
number = {1},
pages = {105-117},
publisher = {Société mathématique de France},
title = {Semistability of Frobenius direct images over curves},
url = {http://eudml.org/doc/272383},
volume = {135},
year = {2007},
}

TY - JOUR
AU - Mehta, Vikram B.
AU - Pauly, Christian
TI - Semistability of Frobenius direct images over curves
JO - Bulletin de la Société Mathématique de France
PY - 2007
PB - Société mathématique de France
VL - 135
IS - 1
SP - 105
EP - 117
AB - Let $X$ be a smooth projective curve of genus $g \ge 2$ defined over an algebraically closed field $k$ of characteristic $p&gt;0$. 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 $X$.
LA - eng
KW - vector bundle; semistability; Frobenius
UR - http://eudml.org/doc/272383
ER -

References

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  1. [1] A. Beauville – « On the stability of the direct image of a generic vector bundle », preprint available at http://math.unice.fr/~beauvill/pubs/imdir.pdf. 
  2. [2] G. Faltings – « Projective connections and G -bundles », J. Algebraic Geom.2 (1993), p. 507–568. Zbl0790.14019MR1211997
  3. [3] K. Joshi, S. Ramanan, E. Z. Xia & J. K. Yu – « On vector bundles destabilized by Frobenius pull-back », Compos. Math.142 (2006), p. 616–630. Zbl1101.14049MR2231194
  4. [4] H. Lange & C. Pauly – « On Frobenius-destabilized rank- 2 vector bundles over curves », Comm. Math. Helvetici83 (2008), p. 179–209. Zbl1157.14017MR2365412
  5. [5] Y. Laszlo & C. Pauly – « The Frobenius map, rank 2 vector bundles and Kummer’s quartic surface in characteristic 2 and 3 », Adv. Math.185 (2004), p. 246–269. Zbl1055.14038MR2060469
  6. [6] J. Le Potier – « Module des fibrés semi-stables et fonctions thêta », in Moduli of vector bundles (Sanda 1994, Kyoto 1994), Lect. Notes Pure Appl. Math., vol. 179, Dekker, New York, 1996, p. 83–101. Zbl0890.14017MR1397983
  7. [7] V. B. Mehta & S. Subramanian – « Nef line bundles which are not ample », Math. Z.219 (1995), p. 235–244. Zbl0826.14009MR1337219
  8. [8] B. Osserman – « The generalized Verschiebung map for curves of genus 2 », Math. Ann.336 (2006), p. 963–986. Zbl1111.14031MR2255181
  9. [9] M. Raynaud – « Sections des fibrés vectoriels sur une courbe », Bull. Soc. Math. France110 (1982), p. 103–125. Zbl0505.14011MR662131
  10. [10] N. I. Shepherd-Barron – « Semistability and reduction mod p », Topology37 (1998), p. 659–664. Zbl0926.14021MR1604907
  11. [11] X. Sun – « Remarks on semistability of G -bundles in positive characteristic », Compos. Math.119 (1999), p. 41–52. Zbl0951.14031MR1711507

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