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A holomorphic Lagrangian fibration on a holomorphically symplectic manifoldis a holomorphic map with Lagrangian fibers. It is known that a given compactmanifold admits only finitely many holomorphic symplectic structures, up todeformation. We prove that a given compact manifold with $b_2 \geq 7$ admitsonly finitely many deformation types of holomorphic Lagrangian fibrations. Wealso prove that all known hyperkahler manifolds are never Kobayashi hyperbolic.

Families of Lagrangian fibrations on hyperkaehler manifolds