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

By Jian Song, Xiaowei Wang. 2012

86 citations recorded by the source corpus.

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

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/99ea739cf6279f325d36dd4a30cf9ca7dc360cf8

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In Applied Mathematics

In Geometry and Topology

Citations in the corpus

A variational approach to the Yau–Tian–Donaldson conjecture (2015, methodology): “Closely related results were obtained in [LS14, SW16] when supp Δ is a smooth divisor, and in [LTW17] in the general case (building on the preprint version [BBJ15] of the present paper).”