Open Access
February 2010 Adiabatic limits of Ricci-flat Kähler metrics
Valentino Tosatti
J. Differential Geom. 84(2): 427-453 (February 2010). DOI: 10.4310/jdg/1274707320


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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Valentino Tosatti. "Adiabatic limits of Ricci-flat Kähler metrics." J. Differential Geom. 84 (2) 427 - 453, February 2010.


Published: February 2010
First available in Project Euclid: 24 May 2010

zbMATH: 1208.32024
MathSciNet: MR2652468
Digital Object Identifier: 10.4310/jdg/1274707320

Rights: Copyright © 2010 Lehigh University

Vol.84 • No. 2 • February 2010
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