Adiabatic limits of Ricci-flat Kahler metrics

Valentino Tosatti Harvard University

Differential Geometry mathscidoc:1609.10168

Journal of Differential Geometry, 84, (2), 427-453, 2010
We study adiabatic limits of Ricci-flat K¨ahler metrics on a Calabi-Yau manifold which is the total space of a holomorphic fibration when the volume of the fibers goes to zero. By establishing some new a priori estimates for the relevant complex Monge-Amp`ere equation, we show that the Ricci-flat metrics collapse (away from the singular fibers) to a metric on the base of the fibration. This metric has Ricci curvature equal to a WeilPetersson metric that measures the variation of complex structure of the Calabi-Yau fibers. This generalizes results of Gross-Wilson for K3 surfaces to higher dimensions.
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@inproceedings{valentino2010adiabatic,
  title={Adiabatic limits of Ricci-flat Kahler metrics},
  author={Valentino Tosatti},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160911212738342305825},
  booktitle={Journal of Differential Geometry},
  volume={84},
  number={2},
  pages={427-453},
  year={2010},
}
Valentino Tosatti. Adiabatic limits of Ricci-flat Kahler metrics. 2010. Vol. 84. In Journal of Differential Geometry. pp.427-453. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160911212738342305825.
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