Lagrangian fibrations on hyperkähler manifolds - On a question of Beauville

Daniel Greb, Christian Lehn, Sönke Rollenske

Published 2011-05-17, updated 2012-11-08Version 3

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.