Adiabatic limits of Ricci-flat Kahler metrics
Abstract
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.
- Publication:
-
arXiv e-prints
- Pub Date:
- May 2009
- DOI:
- 10.48550/arXiv.0905.4718
- arXiv:
- arXiv:0905.4718
- Bibcode:
- 2009arXiv0905.4718T
- Keywords:
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- Mathematics - Differential Geometry;
- 32Q25;
- 14J32;
- 32Q20;
- 53C25
- E-Print:
- 26 pages