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

Song, Jian; Wang, Xiaowei

doi:10.2140/gt.2016.20.49

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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

DOI: 10.2140/gt.2016.20.49

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 $D \in | - K_{X} |$ is a smooth divisor and the Mabuchi $K$ –energy is bounded below, then there exists a unique conical Kähler–Einstein metric satisfying $Ric (g) = β g + (1 - β) [D]$ for any $β \in (0, 1)$ . We also construct unique conical toric Kähler–Einstein metrics with $β = R (X)$ 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 $R (X) = 1$ .

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

Volume 20, issue 1 (2016)