Greatest lower bounds on the Ricci curvature of Fano manifolds

TL;DR

This paper investigates the maximum lower bound of Ricci curvature achievable on Fano manifolds and relates it to the existence time of a continuity method for Kähler-Einstein metrics, providing explicit results for specific cases.

Contribution

It establishes a link between Ricci curvature bounds and the continuity method, and computes the supremum for certain Fano manifolds like P^2 blown up at one point.

Findings

Supremum of Ricci lower bounds on P^2 blown up at one point is 6/7.

The supremum equals the maximum existence time of Aubin's continuity path.

Upper bounds are provided for other Fano manifolds.

Abstract

On a Fano manifold M we study the supremum of the possible t such that there is a K\"ahler metric in c_1(M) with Ricci curvature bounded below by t. This is shown to be the same as the maximum existence time of Aubin's continuity path for finding K\"ahler-Einstein metrics. We show that on P^2 blown up in one point this supremum is 6/7, and we give upper bounds for other manifolds.