Gromov-Hausdorff limits of collapsing Calabi-Yau fibrations

We study Calabi-Yau metrics on a projective manifold in Kähler classes converging to a semiample class given by a fibration. We show that the Gromov-Hausdorff limit of the metrics is homeomorphic to the base of the fibration and in addition the discriminant locus has Hausdorff codimension at least 2. This resolves conjectures of Tosatti.