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Gromov-Hausdorff collapsing of Calabi-Yau manifolds

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arxiv 1304.1820 v1 pith:LPZJPIF6 submitted 2013-04-05 math.DG math.AGmath.MG

Gromov-Hausdorff collapsing of Calabi-Yau manifolds

classification math.DG math.AGmath.MG
keywords manifoldsarxivcalabi-yaufiberedmetricsahlerbasecollapsing
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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.

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