Gromov-Hausdorff Collapsing of Calabi-Yau manifolds

Abstract

This paper is a sequel to Collapsing of Abelian Fibered Calabi-Yau Manifolds [12]. We further study Gromov–Hausdorff collapsing limits of Ricci-flat Kähler metrics on abelian fibered Calabi–Yau manifolds. Firstly, we show that in the same setup as Collapsing of Abelian Fibered Calabi-Yau Manifolds, 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 Kähler metrics. By combining these two results, we extend Large complex structure limits of K3 surfaces [13] to any fibered projective K3 surface without any assumption on the type of singular fibers.