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2011

Lagrangian fibrations on hyperk\"ahler fourfolds

Greb, Daniel, Lehn, Christian, Rollenske, Sönke

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Let X be a projective hyperk\"ahler manifold containing a Lagrangian subtorus L.

We study intersections of deformations of L, defining a canonical almost holomorphic map called L-reduction, which is not birational if and only if X admits an almost holomorphic Lagrangian fibration with (strong) fibre L.
In dimension four we prove that in the above situation there is always a holomorphic Lagrangian fibration with fibre L, thus answering a question of Beauville in this particular case.

Lagrangian fibrations on hyperk\"ahler fourfolds · Around

Built on

On fibre space structures of a projective irreducible symplectic manifold
Daisuke Matsushita · 1999
Earlier work this paper cites.
Equidimensionality of Lagrangian fibrations on holomorphic symplectic manifolds
Daisuke Matsushita · 2000
Earlier work this paper cites.
Higher-Dimensional Algebraic Geometry
Olivier Debarre · 2001
Earlier work this paper cites.
Addendum: “On fibre space structures of a projective irreducible symplectic manifold” [Topology 38
Daisuke Matsushita · 2001
Earlier work this paper cites.

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Then

Ekaterina Amerik. A remark on a question of Beauville about lagrangian fibrations, arXiv:1110.2852, 2011
Original
2011
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Holomorphic symplectic geometry: a problem list
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Arnaud Beauville · 2011
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Lagrangian fibrations on hyperkähler manifolds — On a question of Beauville
Original
Daniel Greb, Christian Lehn, and Sönke Rollenske · 2011
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