Collapsing of abelian fibred Calabi-Yau manifolds

TL;DR

This paper investigates how Ricci-flat Kähler metrics on abelian fibered Calabi-Yau manifolds collapse as fiber volume shrinks, revealing geometric limits and applications to hyperkähler degenerations and mirror symmetry.

Contribution

It provides a detailed analysis of metric collapse behavior on abelian fibered Calabi-Yau manifolds and extends results to hyperkähler degenerations, connecting to mirror symmetry.

Findings

Metrics collapse with bounded curvature away from critical locus

Rescaled fiber metrics become flat in the limit

Limit metric on the base relates to Gromov-Hausdorff limits

Abstract

We study the collapsing behaviour of Ricci-flat Kahler metrics on a projective Calabi-Yau manifold which admits an abelian fibration, when the volume of the fibers approaches zero. We show that away from the critical locus of the fibration the metrics collapse with locally bounded curvature, and along the fibers the rescaled metrics become flat in the limit. The limit metric on the base minus the critical locus is locally isometric to an open dense subset of any Gromov-Hausdorff limit space of the Ricci-flat metrics. We then apply these results to study metric degenerations of families of polarized hyperkahler manifolds in the large complex structure limit. In this setting we prove an analog of a result of Gross-Wilson for K3 surfaces, which is motivated by the Strominger-Yau-Zaslow picture of mirror symmetry.