Kähler-Einstein metrics on Fano manifolds. II: Limits with cone angle less than \boldmath2𝜋

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

paper · pdf · doi:10.1090/s0894-0347-2014-00800-6

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

Abstract

This is the second of a series of three papers 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 is less than 2 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="pi"> <mml:semantics> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi> π </mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">π </mml:annotation> </mml:semantics> </mml:math> </inline-formula> . We show that these are in a natrual way projective algebraic varieties. In the case when the limiting variety and the limiting divisor are smooth we show that the limiting metric also has standard cone singularities.

Citations

• Lower Bounds on Ricci Curvature and the Almost Rigidity of Warped Products
• Kahler-Einstein metrics on Fano manifolds, I: approximation of metrics with cone singularities
• Gromov–Hausdorff limits of Kähler manifolds and algebraic geometry
• Convergence and rigidity of manifolds under Ricci curvature bounds
• Kähler–Einstein Metrics and Stability

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
• Generalized Matsushima’s theorem and Kähler–Einstein cone metrics
• Simon Donaldson [wikipedia]

Related

• 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, II: limits with cone angle less than 2 π
• 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. I: Approximation of metrics with cone singularities
• Kahler-Einstein metrics on Fano manifolds, I: approximation of metrics with cone singularities