The greatest Ricci lower bound, conical Einstein metrics and the Chern number inequality
Abstract
We partially confirm a conjecture of Donaldson relating the greatest Ricci lower bound
R(X)
to the existence of conical Kahler-Einstein metrics on a Fano manifold
X
. In particular, if
D∈|−
K
X
|
is a smooth simple divisor and the Mabuchi
K
-energy is bounded below, then there exists a unique conical Kahler-Einstein metric satisfying
Ric(g)=βg+(1−β)[D]
for any
β∈(0,1)
. We also construct unique smooth conical toric Kahler-Einstein metrics with
β=R(X)
and a unique effective Q-divisor
D∈[−
K
X
]
for all toric Fano manifolds. Finally we prove a Miyaoka-Yau type inequality for Fano manifolds with
R(X)=1
.