Families of Lagrangian fibrations on hyperkähler manifolds

A holomorphic Lagrangian fibration on a holomorphically symplectic manifold is a holomorphic map with Lagrangian fibers. It is known (due to Huybrechts) that a given compact manifold admits only finitely many holomorphic symplectic structures, up to deformation. We prove that a given compact, simple hyperkähler manifold with b2⩾7 admits only finitely many deformation types of holomorphic Lagrangian fibrations. We also prove that all known hyperkähler manifolds are never Kobayashi hyperbolic.