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ISSN (electronic): 1364-0380
ISSN (print): 1465-3060
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(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.

Keywords
Kähler–Einstein metric, conic Kähler metric, toric variety
Mathematical Subject Classification 2010
Primary: 32Q20, 53C55
References
Publication
Received: 5 December 2013
Revised: 16 April 2015
Accepted: 9 June 2015
Published: 29 February 2016
Proposed: Simon Donaldson
Seconded: John Lott, Bruce Kleiner
Authors
Jian Song
Department of Mathematics
Rutgers University
Piscataway, NJ 08854-8019
USA
Xiaowei Wang
Department of Mathematics and Computer Sciences
Rutgers University
Newark, NJ 07102
USA