Finiteness of Lagrangian fibrations with fixed invariants

We prove finiteness of hyperkaehler Lagrangian fibrations in any fixed dimension with fixed Fujiki constant and discriminant of the Beauville-Bogomolov-Fujiki lattice, up to deformation. We also prove finiteness of hyperk"ahler Lagrangian fibrations with an ample line bundle of a given degree on the general fiber of the fibration.