Kähler-Einstein metrics on Fano manifolds. I: Approximation of metrics with cone singularities

2014/03/28 by Xiuxiong Chen, Simon Donaldson, Song Sun · 5 citations
Mathematics · #Geometry and complex manifolds #Geometric Analysis and Curvature Flows #Fano plane #Mathematics #Gravitational singularity #Divisor (algebraic geometry) #Manifold (fluid mechanics) #Einstein #Metric (unit) #Cone (formal languages) #Pure mathematics #Series (stratigraphy) #Hyperkähler manifold #Mathematical analysis #Geometry #Mathematical physics #Ricci-flat manifold #Curvature #Scalar curvature

paper · pdf · doi:10.1090/s0894-0347-2014-00799-2

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

Abstract

This is the first of a series of three papers which prove the fact that a <italic>K</italic> -stable Fano manifold admits a Kähler-Einstein metric. The main result of this paper is that a Kähler-Einstein metric with cone singularities along a divisor can be approximated by a sequence of smooth Kähler metrics with controlled geometry in the Gromov-Hausdorff sense.

Citations

• Elliptic Partial Differential Equations of Second Order
• A Brunn-Minkowski type inequality for Fano manifolds and the Bando-Mabuchi uniqueness theorem
• On the regularity of the monge‐ampère equation det (∂ 2 u/∂x i ∂x j ) = f(x, u)
• Improper affine hyperspheres of convex type and a generalization of a theorem by K. Jörgens.
• On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I
• Complete Kähler manifolds with zero Ricci curvature. I
• Complete Kahler Manifolds with Zero Ricci Curvature. I

Cited by

• On K-polystability for log del Pezzo pairs of Maeda type
• Kähler–Einstein Metrics on Smooth Fano Symmetric Varieties with Picard Number One
• C^2,α-estimate for conical Kähler-Ricci flow
• Volume and Monge–Ampère energy on polarized affine varieties
• Simon Donaldson [wikipedia]

Related

• Kahler-Einstein metrics on Fano manifolds, I: approximation of metrics with cone singularities
• Kähler-Einstein metrics on Fano manifolds. III: Limits as cone angle approaches \boldmath2𝜋 and completion of the main proof
• 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
• Kähler-Einstein metrics along the smooth continuity method