Duke Mathematical Journal

Collapsing of abelian fibered Calabi–Yau manifolds

Mark Gross, Valentino Tosatti, and Yuguang Zhang

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Abstract

We study the collapsing behavior of Ricci-flat Kähler metrics on a projective Calabi–Yau manifold which admits an abelian fibration, when the volume of the fibers approaches zero. We show that away from the critical locus of the fibration the metrics collapse with locally bounded curvature, and along the fibers the rescaled metrics become flat in the limit. The limit metric on the base minus the critical locus is locally isometric to an open dense subset of any Gromov–Hausdorff limit space of the Ricci-flat metrics. We then apply these results to study metric degenerations of families of polarized hyperkähler manifolds in the large complex structure limit. In this setting, we prove an analogue of a result of Gross and Wilson for K3 surfaces, which is motivated by the Strominger–Yau–Zaslow picture of mirror symmetry.

Article information

Source
Duke Math. J., Volume 162, Number 3 (2013), 517-551.

Dates
First available in Project Euclid: 14 February 2013

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1360874854

Digital Object Identifier
doi:10.1215/00127094-2019703

Mathematical Reviews number (MathSciNet)
MR3024092

Zentralblatt MATH identifier
1276.32020

Subjects
Primary: 32Q25: Calabi-Yau theory [See also 14J30]
Secondary: 32W20: Complex Monge-Ampère operators 14J32: Calabi-Yau manifolds 14J33: Mirror symmetry [See also 11G42, 53D37] 53C26: Hyper-Kähler and quaternionic Kähler geometry, "special" geometry

Citation

Gross, Mark; Tosatti, Valentino; Zhang, Yuguang. Collapsing of abelian fibered Calabi–Yau manifolds. Duke Math. J. 162 (2013), no. 3, 517--551. doi:10.1215/00127094-2019703. https://projecteuclid.org/euclid.dmj/1360874854


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