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

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 $R(X)=1$.