Kähler-Einstein metrics on Fano manifolds. III: Limits as cone angle approaches \boldmath2𝜋 and completion of the main proof

2014/03/28 by Xiuxiong Chen, Simon Donaldson, Song Sun · 5 citations
Mathematics · #Geometry and complex manifolds #Geometric Analysis and Curvature Flows #Algebraic Geometry and Number Theory #Fano plane #Mathematics #Manifold (fluid mechanics) #Gravitational singularity #Cone (formal languages) #Limiting #Metric (unit) #Pure mathematics #Functor #Algorithm #Mathematical analysis

paper · pdf · doi:10.1090/s0894-0347-2014-00801-8

openalex publication_date 2014/03/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/14

Abstract

This is the third and final article in a series which prove the fact that a <italic>K</italic> -stable Fano manifold admits a Kähler-Einstein metric. In this paper we consider the Gromov-Hausdorff limits of metrics with cone singularities in the case when the limiting cone angle approaches 2 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="pi"> <mml:semantics> <mml:mi> π </mml:mi> <mml:annotation encoding="application/x-tex">π</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . We also put all our technical results together to complete the proof of the main theorem.

Citations

• Scalar Curvature and Stability of Toric Varieties
• Positivity in Algebraic Geometry I
• Elliptic Partial Differential Equations of Second Order
• A Brunn-Minkowski type inequality for Fano manifolds and the Bando-Mabuchi uniqueness theorem
• Extremal Kähler Metrics II
• On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I
• Kahler-Einstein metrics on Fano manifolds, I: approximation of metrics with cone singularities
• Introduction to Complex Analysis in Several Variables
• Ricci flow with surgery on three-manifolds
• Lectures on Hermitian-Einstein Metrics for Stable Bundles and Kähler-Einstein Metrics: Delivered at the German Mathematical Society Seminar in Düsseldorf in June, 1986
• On Kähler Manifolds with Vanishing Canonical Class
• Kahler-Einstein metrics on Fano manifolds, II: limits with cone angle less than 2 π

Cited by

• On K-polystability for log del Pezzo pairs of Maeda type
• ON THE MODIFIED FUTAKI INVARIANT OF COMPLETE INTERSECTIONS IN PROJECTIVE SPACES
• Kähler–Einstein Metrics on Smooth Fano Symmetric Varieties with Picard Number One
• C^2,α-estimate for conical Kähler-Ricci flow
• Simon Donaldson [wikipedia]

Related

• Kahler-Einstein metrics on Fano manifolds, III: limits as cone angle approaches 2π and completion of the main proof
• Kähler-Einstein metrics on Fano manifolds. II: Limits with cone angle less than \boldmath2𝜋
• Kahler-Einstein metrics on Fano manifolds, II: limits with cone angle less than 2 π
• Kähler-Einstein metrics on Fano manifolds. I: Approximation of metrics with cone singularities
• Kahler-Einstein metrics on Fano manifolds, I: approximation of metrics with cone singularities