## Geometry & Topology

### 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 $R(X)$ to the existence of conical Kähler–Einstein metrics on a Fano manifold $X$. In particular, if $D ∈|− KX|$ is a smooth divisor and the Mabuchi $K$–energy is bounded below, then there exists a unique conical Kähler–Einstein metric satisfying $Ric(g) = βg + (1 − β)[D]$ for any $β ∈ (0,1)$. We also construct unique conical toric Kähler–Einstein metrics with $β = R(X)$ and a unique effective $ℚ$–divisor $D ∈ [−KX]$ for all toric Fano manifolds. Finally we prove a Miyaoka–Yau-type inequality for Fano manifolds with $R(X) = 1$.

#### Article information

Source
Geom. Topol., Volume 20, Number 1 (2016), 49-102.

Dates
Revised: 16 April 2015
Accepted: 9 June 2015
First available in Project Euclid: 16 November 2017

https://projecteuclid.org/euclid.gt/1510858923

Digital Object Identifier
doi:10.2140/gt.2016.20.49

Mathematical Reviews number (MathSciNet)
MR3470713

Zentralblatt MATH identifier
1342.32018

#### Citation

Song, Jian; Wang, Xiaowei. The greatest Ricci lower bound, conical Einstein metrics and Chern number inequality. Geom. Topol. 20 (2016), no. 1, 49--102. doi:10.2140/gt.2016.20.49. https://projecteuclid.org/euclid.gt/1510858923

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