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

TL;DR

This paper explores the relationship between the greatest Ricci lower bound and conical Kähler-Einstein metrics on Fano manifolds, confirming conjectures and establishing existence and uniqueness results.

Contribution

It partially confirms Donaldson's conjecture linking Ricci bounds to conical Einstein metrics and constructs unique metrics for toric Fano manifolds.

Findings

Existence of conical Kähler-Einstein metrics under certain conditions.

Construction of unique smooth conical toric Kähler-Einstein metrics.

Proof of a Miyaoka-Yau type inequality for Fano manifolds with Ricci bound 1.

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 $R(X)=1$.