Kahler-Einstein metrics on Fano manifolds, III: limits as cone angle approaches 2\pi\ and completion of the main proof

TL;DR

This paper completes the proof that K-stable Fano manifolds admit Kähler-Einstein metrics by analyzing limits of cone singularity metrics as the cone angle approaches 2π, consolidating previous technical results.

Contribution

It finalizes the proof of existence of Kähler-Einstein metrics on K-stable Fano manifolds by studying Gromov-Hausdorff limits of cone metrics.

Findings

Gromov-Hausdorff limits of cone metrics as angle approaches 2π analyzed

Main theorem on existence of Kähler-Einstein metrics on K-stable Fano manifolds proved

Technical results integrated to complete the proof

Abstract

This is the third and final paper in a series which establish results announced in arXiv:1210.7494. In this paper we consider the Gromov-Hausdorff limits of metrics with cone singularities in the case when the limiting cone angle approaches 2\pi. We also put all our technical results together to complete the proof of the main theorem that if a K-stable Fano manifold admits a Kahler-Einstein metric.