Geometry & Topology

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

Jian Song and Xiaowei Wang

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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 –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

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

Received: 5 December 2013
Revised: 16 April 2015
Accepted: 9 June 2015
First available in Project Euclid: 16 November 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 32Q20: Kähler-Einstein manifolds [See also 53Cxx] 53C55: Hermitian and Kählerian manifolds [See also 32Cxx]

Kähler–Einstein metric conic Kähler metric toric variety


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.

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