Adiabatic limits of Ricci-flat Kahler metrics

TL;DR

This paper investigates the behavior of Ricci-flat Kähler metrics on Calabi-Yau manifolds under fiber volume collapse, revealing convergence to a base metric with Weil-Petersson curvature, extending known results to higher dimensions.

Contribution

It establishes new a priori estimates for complex Monge-Ampère equations and generalizes the collapse results from K3 surfaces to higher-dimensional Calabi-Yau manifolds.

Findings

Ricci-flat metrics collapse to a base metric with Weil-Petersson curvature

Established new a priori estimates for complex Monge-Ampère equations

Extended collapse results from K3 surfaces to higher dimensions

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.