On Deformations of Lagrangian Fibrations

Abstract

Let X be an irreducible symplectic manifold and Def(X) be the Kuranishi family. Assume that X admits a Lagrangian fibration. We prove that X can be deformed preserving a Lagrangian fibration. More precisely, there exists a smooth hypersurface H of Def(X), such that the restriction family χ ×Def(X) H admits a family of Lagrangian fibrations over H.