Gromov-Hausdorff collapsing of Calabi-Yau manifolds

TL;DR

This paper investigates the limits of Ricci-flat Kähler metrics on abelian fibered Calabi-Yau manifolds under Gromov-Hausdorff collapse, revealing topological and geometric properties of the limit spaces.

Contribution

It extends previous work by showing the limit space is homeomorphic to the base when the base is one-dimensional and identifies special Kähler structures in certain fibrations, broadening understanding of Calabi-Yau degenerations.

Findings

Limit space homeomorphic to base manifold for one-dimensional base

Regular parts of limits are special Kähler metrics in Lagrangian fibrations

Generalizes results to any fibered projective K3 surface without singular fiber restrictions

Abstract

This paper is a sequel to arXiv:1108.0967. We further study Gromov-Hausdorff collapsing limits of Ricci-flat K\"ahler metrics on abelian fibered Calabi-Yau manifolds. Firstly, we show that in the same setup as arXiv:1108.0967, if the dimension of the base manifold is one, the limit metric space is homeomorphic to the base manifold. Secondly, if the fibered Calabi-Yau manifolds are Lagrangian fibrations of holomorphic symplectic manifolds, the metrics on the regular parts of the limits are special K\"ahler metrics. By combining these two results, we extend arXiv:math/0008018 to any fibered projective K3 surface without any assumption on the type of singular fibers.