On coverings of simple abelian varieties

Olivier Debarre

Bulletin de la Société Mathématique de France (2006)

  • Volume: 134, Issue: 2, page 253-260
  • ISSN: 0037-9484

Abstract

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To any finite covering f : Y → X of degree d between smooth complex projective manifolds, one associates a vector bundle E f of rank d - 1 on X whose total space contains Y . It is known that E f is ample when X is a projective space ([Lazarsfeld 1980]), a Grassmannian ([Manivel 1997]), or a Lagrangian Grassmannian ([Kim Maniel 1999]). We show an analogous result when X is a simple abelian variety and f does not factor through any nontrivial isogeny X ' → X . This result is obtained by showing that E f is M -regular in the sense of Pareschi-Popa, and that any M -regular sheaf is ample.

How to cite

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Debarre, Olivier. "On coverings of simple abelian varieties." Bulletin de la Société Mathématique de France 134.2 (2006): 253-260. <http://eudml.org/doc/272380>.

@article{Debarre2006,
abstract = {To any finite covering $f:Y\rightarrow X$ of degree $d$ between smooth complex projective manifolds, one associates a vector bundle $E_f$ of rank $d-1$ on $X$ whose total space contains $Y$. It is known that $E_f$ is ample when $X$ is a projective space ([Lazarsfeld 1980]), a Grassmannian ([Manivel 1997]), or a Lagrangian Grassmannian ([Kim Maniel 1999]). We show an analogous result when $X$ is a simple abelian variety and $f$ does not factor through any nontrivial isogeny $X^\{\prime \}\rightarrow X$. This result is obtained by showing that $E_f$ is $M$-regular in the sense of Pareschi-Popa, and that any $M$-regular sheaf is ample.},
author = {Debarre, Olivier},
journal = {Bulletin de la Société Mathématique de France},
keywords = {abelian variety; vector bundle; ample sheaf; $M$-regular sheaf; continuously generated sheaf; Barth-Lefschetz theorem; Mukai transform},
language = {eng},
number = {2},
pages = {253-260},
publisher = {Société mathématique de France},
title = {On coverings of simple abelian varieties},
url = {http://eudml.org/doc/272380},
volume = {134},
year = {2006},
}

TY - JOUR
AU - Debarre, Olivier
TI - On coverings of simple abelian varieties
JO - Bulletin de la Société Mathématique de France
PY - 2006
PB - Société mathématique de France
VL - 134
IS - 2
SP - 253
EP - 260
AB - To any finite covering $f:Y\rightarrow X$ of degree $d$ between smooth complex projective manifolds, one associates a vector bundle $E_f$ of rank $d-1$ on $X$ whose total space contains $Y$. It is known that $E_f$ is ample when $X$ is a projective space ([Lazarsfeld 1980]), a Grassmannian ([Manivel 1997]), or a Lagrangian Grassmannian ([Kim Maniel 1999]). We show an analogous result when $X$ is a simple abelian variety and $f$ does not factor through any nontrivial isogeny $X^{\prime }\rightarrow X$. This result is obtained by showing that $E_f$ is $M$-regular in the sense of Pareschi-Popa, and that any $M$-regular sheaf is ample.
LA - eng
KW - abelian variety; vector bundle; ample sheaf; $M$-regular sheaf; continuously generated sheaf; Barth-Lefschetz theorem; Mukai transform
UR - http://eudml.org/doc/272380
ER -

References

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  2. [2] —, « Tores et variétés abéliennes complexes », Cours spécialisés, vol. 6, Société Math. France, Paris ; EDP Sciences, Les Ulis, 1999. Zbl0964.14037
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  7. [7] M. Kim & L. Manivel – « On branched coverings of some homogeneous spaces », Topology38 (1999), p. 1141–1160. Zbl0935.14008MR1688418
  8. [8] K. Kubota – « Ample sheaves », J. Fac. Sci. Univ. Tokyo Sect. I A Math.17 (1970), p. 421–430. Zbl0212.26102MR292849
  9. [9] R. Lazarsfeld – « A Barth-type theorem for branched coverings of projective space », Math. Ann.249 (1980), p. 153–162. Zbl0434.32013MR578722
  10. [10] —, « Positivity in algebraic geometry II », Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 49, Springer-Verlag, Heidelberg, 2004. Zbl0633.14016MR2095472
  11. [11] L. Manivel – « Vanishing theorems for ample vector bundles », Invent. Math.127 (1997), p. 401–416. Zbl0906.14011MR1427625
  12. [12] S. Mukai – « Duality between D ( X ) and D ( X ^ ) with its application to Picard sheaves », Nagoya Math. J.81 (1981), p. 153–175. Zbl0417.14036MR607081
  13. [13] G. Pareschi & M. Popa – « Regularity on abelian varieties I », J. Amer. Math. Soc.16 (2003), p. 285–302. Zbl1022.14012MR1949161
  14. [14] T. Peternell & A. Sommese – « Ample Vector Bundles and Branched Coverings, II », The Fano Conference (A. Collino, A. Conte & M. Marchiso, éds.), Univ. Torino, 2004, p. 625–645. Zbl1071.14018MR2112595
  15. [15] C. Simpson – « Subspaces of moduli spaces of rank one local systems », Ann. Sci. École Norm. Sup.26 (1993), p. 361–401. Zbl0798.14005MR1222278

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