Adiabatic limits of Ricci-flat Kahler metrics

Details Adiabatic limits of Ricci-flat Kahler metrics Adiabatic limits of Ricci-flat Kahler metrics

2009-05-28

arxiv.org/abs/0905.4718v2

J. Differential Geom. 84 (2010), no.2, 427-453

Mathematics

Differential Geometry

26 pages; final version to appear in J. Differential Geom

Scientific paper

We study adiabatic limits of Ricci-flat Kahler 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-Ampere 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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