15 February 2013 Collapsing of abelian fibered Calabi–Yau manifolds
Mark Gross, Valentino Tosatti, Yuguang Zhang
Duke Math. J. 162(3): 517-551 (15 February 2013). DOI: 10.1215/00127094-2019703


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.


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Mark Gross. Valentino Tosatti. Yuguang Zhang. "Collapsing of abelian fibered Calabi–Yau manifolds." Duke Math. J. 162 (3) 517 - 551, 15 February 2013. https://doi.org/10.1215/00127094-2019703


Published: 15 February 2013
First available in Project Euclid: 14 February 2013

zbMATH: 1276.32020
MathSciNet: MR3024092
Digital Object Identifier: 10.1215/00127094-2019703

Primary: 32Q25
Secondary: 14J32 , 14J33 , 32W20 , 53C26

Rights: Copyright © 2013 Duke University Press


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Vol.162 • No. 3 • 15 February 2013
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