A semiampleness result for hyperkähler fourfolds and smoothings of Lagrangian subvarieties

We give a geometric proof of the following statement: let $X$ be a smooth hyperkähler fourfold such that the Picard group is generated by a line bundle $L$ that is nef, non-trivial, and not big. If the Kodaira dimension of $L$ is non-negative, some positive multiple defines a Lagrangian fibration. The proof requires a number of tools which are of independent interest, including an analogue of Voisin’s unobstructedness result and a smoothing result for singular Lagrangian subvarieties.