On deformations of Lagrangian fibrations

Understand

Let X be an irreducible symplectic manifold and Def(X) the Kuranishi space.

• 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 over H admits a family of Lagrangian fibrations over H.