A uniform L^{\infty} estimate for complex Monge-Ampere equations

TL;DR

This paper establishes uniform sup-norm bounds for complex Monge-Ampère equations under degenerating Kähler metrics and applies these results to demonstrate the existence of generalized Kähler-Einstein metrics as limits of the Kähler-Ricci flow in certain fibrations.

Contribution

It provides a new uniform estimate for Monge-Ampère equations with degenerating metrics and uses this to prove the existence of generalized Kähler-Einstein metrics via the Kähler-Ricci flow.

Findings

Uniform sup-norm estimates for Monge-Ampère equations with degenerating metrics

Existence of generalized Kähler-Einstein metrics as limits of Kähler-Ricci flow

Application to holomorphic fibrations and degenerating Kähler metrics

Abstract

We prove uniform sup-norm estimates for the Monge-Ampere equation with respect to a family of Kahler metrics which degenerate towards a pull-back of a metric from a lower dimensional manifold. This is then used to show the existence of generalized Kahler-Einstein metrics as the limits of the Kahler-Ricci flow for some holomorphic fibrations (in the spirit of Song and Tian "The Kahler-Ricci flow on surfaces of positive Kodaira dimension", arXiv:math/0602150).