The hyper-Kähler SYZ deformation conjecture

A hyper-Kähler manifold is a simply connected smooth compact Kähler manifold carrying a nowhere-degenerate holomorphic $2$-form. A Lagrangian fibration is a fibration whose fibers are Lagrangian with respect to the holomorphic symplectic form.

Hyper-Kähler SYZ deformation conjecture. Any hyper-Kähler manifold can be deformed into a hyper-Kähler manifold with a Lagrangian fibration.

This is one of the two key conjectures discussed in the paper. It holds for all currently known examples of hyper-Kähler manifolds, but is not established in general.