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

Abstract

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