The greatest Ricci lower bound, conical Einstein metrics and the Chern number inequality

Abstract

We partially confirm a conjecture of Donaldson relating the greatest Ricci lower bound $R(X)$ to the existence of conical Kahler-Einstein metrics on a Fano manifold $X$. In particular, if $D\in |-K_X|$ is a smooth simple divisor and the Mabuchi $K$-energy is bounded below, then there exists a unique conical Kahler-Einstein metric satisfying $Ric(g)=\beta g+(1-\beta )[D]$ for any $\beta \in (0,1)$. We also construct unique smooth conical toric Kahler-Einstein metrics with $\beta =R(X)$ and a unique effective Q-divisor $D\in [-K_X]$ for all toric Fano manifolds. Finally we prove a Miyaoka-Yau type inequality for Fano manifolds with $R(X)=1$.

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• Greatest lower bounds on the Ricci curvature of Fano manifolds
• Curvature, Cones, and Characteristic Numbers
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Cited by in corpus (15)

• On the proper moduli spaces of smoothable Kähler-Einstein Fano varieties
• Existence and deformations of Kahler-Einstein metrics on smoothable Q-Fano varieties
• Basis log canonical thresholds, local intersection estimates, and asymptotically log del Pezzo surfaces
• Mabuchi Solitons and Relative Ding Stability of Toric Fano Varieties
• Orbifold Stability and Miyaoka-Yau Inequality for minimal pairs
• Degenerating Kähler-Einstein cones, locally symmetric cusps, and the Tian-Yau metric
• Scalar curvature and Futaki invariant of Kähler metrics with cone singularities along a divisor
• Construction of constant scalar curvature Kähler cone metrics
• Expansion formula for complex Monge-Ampère equation along cone singularities
• Existence of constant scalar curvature Kaehler cone metrics, properness and geodesic stability
• Singularities of plane complex curves and limits of Kähler metrics with cone singularities. I: Tangent Cones
• Schauder estimates for equations with cone metrics, II
• ALE Calabi-Yau metrics with conical singularities along a compact divisor
• Greatest Ricci lower bounds of projective horospherical manifolds of Picard number one
• Kähler-Einstein metrics on smooth Fano toroidal symmetric varieties of type AIII