Gromov-Hausdorff collapsing of Calabi-Yau manifolds

Mark Gross; Valentino Tosatti; Yuguang Zhang

doi:10.4310/CAG.2016.v24.n1.a4

Gromov-Hausdorff collapsing of Calabi-Yau manifolds

Mark Gross, Valentino Tosatti, Yuguang Zhang

Mathematics

Research output: Contribution to journal › Article › peer-review

32 Scopus citations

Abstract

This paper is a sequel to [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 [12], 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 [13] to any fibered projective K3 surface without any assumption on the type of singular fibers.

Original language	English (US)
Pages (from-to)	93-113
Number of pages	21
Journal	Communications in Analysis and Geometry
Volume	24
Issue number	1
DOIs	https://doi.org/10.4310/CAG.2016.v24.n1.a4
State	Published - 2016

Funding

Supported in part by NSF grant DMS-1105871. Supported in part by a Sloan Research Fellowship and NSF grant DMS-1236969. Supported in part by NSFC-11271015

ASJC Scopus subject areas

Analysis
Statistics and Probability
Geometry and Topology
Statistics, Probability and Uncertainty

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Cite this

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title = "Gromov-Hausdorff collapsing of Calabi-Yau manifolds",

abstract = "This paper is a sequel to [12]. We further study Gromov-Hausdorff collapsing limits of Ricci-flat K{\"a}hler metrics on abelian fibered Calabi-Yau manifolds. Firstly, we show that in the same setup as [12], 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{\"a}hler metrics. By combining these two results, we extend [13] to any fibered projective K3 surface without any assumption on the type of singular fibers.",

author = "Mark Gross and Valentino Tosatti and Yuguang Zhang",

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AB - This paper is a sequel to [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 [12], 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 [13] to any fibered projective K3 surface without any assumption on the type of singular fibers.

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Gromov-Hausdorff collapsing of Calabi-Yau manifolds

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