Lagrangian fibrations on hyperkähler manifolds – On a question of Beauville

Daniel Greb; Christian Lehn; Sönke Rollenske

Annales scientifiques de l'École Normale Supérieure (2013)

  • Volume: 46, Issue: 3, page 375-403
  • ISSN: 0012-9593

Abstract

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Let  X be a compact hyperkähler manifold containing a complex torus L as a Lagrangian subvariety. Beauville posed the question whether X admits a Lagrangian fibration with fibre L . We show that this is indeed the case if X is not projective. If X is projective we find an almost holomorphic Lagrangian fibration with fibre L under additional assumptions on the pair ( X , L ) , which can be formulated in topological or deformation-theoretic terms. Moreover, we show that for any such almost holomorphic Lagrangian fibration there exists a smooth good minimal model, i.e., a hyperkähler manifold birational to  X on which the fibration is holomorphic.

How to cite

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Greb, Daniel, Lehn, Christian, and Rollenske, Sönke. "Lagrangian fibrations on hyperkähler manifolds – On a question of Beauville." Annales scientifiques de l'École Normale Supérieure 46.3 (2013): 375-403. <http://eudml.org/doc/272102>.

@article{Greb2013,
abstract = {Let $X$ be a compact hyperkähler manifold containing a complex torus $L$ as a Lagrangian subvariety. Beauville posed the question whether $X$ admits a Lagrangian fibration with fibre $L$. We show that this is indeed the case if $X$ is not projective. If $X$ is projective we find an almost holomorphic Lagrangian fibration with fibre $L$ under additional assumptions on the pair $(X, L)$, which can be formulated in topological or deformation-theoretic terms. Moreover, we show that for any such almost holomorphic Lagrangian fibration there exists a smooth good minimal model, i.e., a hyperkähler manifold birational to $X$ on which the fibration is holomorphic.},
author = {Greb, Daniel, Lehn, Christian, Rollenske, Sönke},
journal = {Annales scientifiques de l'École Normale Supérieure},
keywords = {hyperkähler manifold; lagrangian fibration},
language = {eng},
number = {3},
pages = {375-403},
publisher = {Société mathématique de France},
title = {Lagrangian fibrations on hyperkähler manifolds – On a question of Beauville},
url = {http://eudml.org/doc/272102},
volume = {46},
year = {2013},
}

TY - JOUR
AU - Greb, Daniel
AU - Lehn, Christian
AU - Rollenske, Sönke
TI - Lagrangian fibrations on hyperkähler manifolds – On a question of Beauville
JO - Annales scientifiques de l'École Normale Supérieure
PY - 2013
PB - Société mathématique de France
VL - 46
IS - 3
SP - 375
EP - 403
AB - Let $X$ be a compact hyperkähler manifold containing a complex torus $L$ as a Lagrangian subvariety. Beauville posed the question whether $X$ admits a Lagrangian fibration with fibre $L$. We show that this is indeed the case if $X$ is not projective. If $X$ is projective we find an almost holomorphic Lagrangian fibration with fibre $L$ under additional assumptions on the pair $(X, L)$, which can be formulated in topological or deformation-theoretic terms. Moreover, we show that for any such almost holomorphic Lagrangian fibration there exists a smooth good minimal model, i.e., a hyperkähler manifold birational to $X$ on which the fibration is holomorphic.
LA - eng
KW - hyperkähler manifold; lagrangian fibration
UR - http://eudml.org/doc/272102
ER -

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