Volume 20, issue 1 (2016)

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

Jian Song and Xiaowei Wang

Geometry & Topology 20 (2016) 49–102
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 $K\phantom{\rule{0.3em}{0ex}}$–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 $ℚ$–divisor $D\in \left[-{K}_{X}\right]$ for all toric Fano manifolds. Finally we prove a Miyaoka–Yau-type inequality for Fano manifolds with $R\left(X\right)=1$.

Keywords
Kähler–Einstein metric, conic Kähler metric, toric variety
Mathematical Subject Classification 2010
Primary: 32Q20, 53C55