## Journal of Differential Geometry

### Adiabatic limits of Ricci-flat Kähler metrics

Valentino Tosatti

#### Abstract

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.

#### Article information

Source
J. Differential Geom., Volume 84, Number 2 (2010), 427-453.

Dates
First available in Project Euclid: 24 May 2010

https://projecteuclid.org/euclid.jdg/1274707320

Digital Object Identifier
doi:10.4310/jdg/1274707320

Mathematical Reviews number (MathSciNet)
MR2652468

Zentralblatt MATH identifier
1208.32024

#### Citation

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