Gromov–Hausdorff collapsing of Calabi–Yau manifolds

Abstract

This paper is a sequel to arXiv:1108.0967. We further study Gromov-Hausdorff\ncollapsing limits of Ricci-flat K\\"ahler metrics on abelian fibered Calabi-Yau\nmanifolds. Firstly, we show that in the same setup as arXiv:1108.0967, if the\ndimension of the base manifold is one, the limit metric space is homeomorphic\nto the base manifold. Secondly, if the fibered Calabi-Yau manifolds are\nLagrangian fibrations of holomorphic symplectic manifolds, the metrics on the\nregular parts of the limits are special K\\"ahler metrics. By combining these\ntwo results, we extend arXiv:math/0008018 to any fibered projective K3 surface\nwithout any assumption on the type of singular fibers.\n