Lagrangian fibrations on generalized Kummer varieties

Martin G. Gulbrandsen

Bulletin de la Société Mathématique de France (2007)

  • Volume: 135, Issue: 2, page 283-298
  • ISSN: 0037-9484

Abstract

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We investigate the existence of Lagrangian fibrations on the generalized Kummer varieties of Beauville. For a principally polarized abelian surface A of Picard number one we find the following: The Kummer variety K n A is birationally equivalent to another irreducible symplectic variety admitting a Lagrangian fibration, if and only if n is a perfect square. And this is the case if and only if K n A carries a divisor with vanishing Beauville-Bogomolov square.

How to cite

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Gulbrandsen, Martin G.. "Lagrangian fibrations on generalized Kummer varieties." Bulletin de la Société Mathématique de France 135.2 (2007): 283-298. <http://eudml.org/doc/272424>.

@article{Gulbrandsen2007,
abstract = {We investigate the existence of Lagrangian fibrations on the generalized Kummer varieties of Beauville. For a principally polarized abelian surface $A$ of Picard number one we find the following: The Kummer variety $K^nA$ is birationally equivalent to another irreducible symplectic variety admitting a Lagrangian fibration, if and only if $n$ is a perfect square. And this is the case if and only if $K^nA$ carries a divisor with vanishing Beauville-Bogomolov square.},
author = {Gulbrandsen, Martin G.},
journal = {Bulletin de la Société Mathématique de France},
keywords = {generalized Kummer varieties; lagrangian fibrations; symplectic varieties},
language = {eng},
number = {2},
pages = {283-298},
publisher = {Société mathématique de France},
title = {Lagrangian fibrations on generalized Kummer varieties},
url = {http://eudml.org/doc/272424},
volume = {135},
year = {2007},
}

TY - JOUR
AU - Gulbrandsen, Martin G.
TI - Lagrangian fibrations on generalized Kummer varieties
JO - Bulletin de la Société Mathématique de France
PY - 2007
PB - Société mathématique de France
VL - 135
IS - 2
SP - 283
EP - 298
AB - We investigate the existence of Lagrangian fibrations on the generalized Kummer varieties of Beauville. For a principally polarized abelian surface $A$ of Picard number one we find the following: The Kummer variety $K^nA$ is birationally equivalent to another irreducible symplectic variety admitting a Lagrangian fibration, if and only if $n$ is a perfect square. And this is the case if and only if $K^nA$ carries a divisor with vanishing Beauville-Bogomolov square.
LA - eng
KW - generalized Kummer varieties; lagrangian fibrations; symplectic varieties
UR - http://eudml.org/doc/272424
ER -

References

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  2. [2] C. Birkenhake & H. Lange – Complex abelian varieties, second éd., Grundlehren der Mathematischen Wissenschaften, vol. 302, Springer, 2004. Zbl1056.14063MR2062673
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  10. [10] —, « Holomorphic symplectic manifolds and Lagrangian fibrations », Acta Appl. Math. 75 (2003), p. 117–123, Monodromy and differential equations (Moscow, 2001). Zbl1030.53075MR1975562
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  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] —, « Fourier functor and its application to the moduli of bundles on an abelian variety », in Algebraic geometry, Sendai, 1985, Adv. Stud. Pure Math., vol. 10, North-Holland, 1987, p. 515–550. Zbl0672.14025MR946249
  14. [14] D. Mumford – Abelian varieties, Tata Institute of Fundamental Research Studies in Mathematics, No. 5, Published for the Tata Institute of Fundamental Research, Bombay, 1970. Zbl0223.14022MR282985
  15. [15] J. Sawon – « Lagrangian fibrations on Hilbert schemes of points on K 3 surfaces », arXiv:math.AG/0509224. Zbl1123.14008MR2306277
  16. [16] K. Yoshioka – « Moduli spaces of stable sheaves on abelian surfaces », Math. Ann.321 (2001), p. 817–884. Zbl1066.14013MR1872531

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