Journal of Differential Geometry

Adiabatic limits of Ricci-flat Kähler metrics

Valentino Tosatti

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We study adiabatic limits of Ricci-flat Kähler 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-Ampère 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 Weil- Petersson 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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J. Differential Geom., Volume 84, Number 2 (2010), 427-453.

First available in Project Euclid: 24 May 2010

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Tosatti, Valentino. Adiabatic limits of Ricci-flat Kähler metrics. J. Differential Geom. 84 (2010), no. 2, 427--453. doi:10.4310/jdg/1274707320.

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