Families of Lagrangian fibrations on hyperkaehler manifolds

Abstract

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

13 pages, v. 2.0, added a section about Kobayashi (non-)hyperbolicity of all known hk manifolds. arXiv admin note: text overlap with arXiv:1008.2480