Lagrangian fibrations on hyperk\"ahler manifolds - On a question of Beauville

Understand

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.