Lagrangian fibrations on hyperkähler manifolds - On a question of Beauville

Details Lagrangian fibrations on hyperkähler manifolds - On a question of Beauville Lagrangian fibrations on hyperkähler manifolds - On a question of Beauville

2011-05-17

arxiv.org/abs/1105.3410v2

Mathematics

Algebraic Geometry

28 pages, v2: minor corrections, added Thm 6.7

Scientific paper

Let X be a compact hyperk\"ahler 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 hyperk\"ahler manifold birational to X on which the fibration is holomorphic.

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