Gromov-Hausdorff limits of collapsing Calabi-Yau fibrations

TL;DR

This paper investigates the limits of Calabi-Yau metrics on projective manifolds as they collapse along fibrations, proving the Gromov-Hausdorff limit is homeomorphic to the base and addressing conjectures about the discriminant locus.

Contribution

It establishes the homeomorphism of the Gromov-Hausdorff limit to the base and shows the discriminant locus has codimension at least 2, resolving conjectures of Tosatti.

Findings

Gromov-Hausdorff limit is homeomorphic to the base of the fibration

Discriminant locus has Hausdorff codimension at least 2

Resolves conjectures of Tosatti regarding collapsing Calabi-Yau metrics

Abstract

We study Calabi-Yau metrics on a projective manifold in K\"ahler 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.