# The Kaehler-Einstein Problem, Tian's $\alpha$-invariant and $\mathbb{C}_+$-actions on affine cones

### Dr A. Wilson (University of Glasgow)

Monday 20th May, 2013 16:00-17:00 Mathematics 325

#### Abstract

I'll talk abou t a link between the existence of a non-trivial $\mathbb{C}_+$-action on the af fi ne cones over (Fano) varieties and their $K$-stability.  Inde ed , for a Fano variety $X$ when Tian's $\alpha$-invariant and its equiva le n t algebraic counterpart, the log canonical threshold, are greater than $(\dim X)/(\dim X + 1)$, $X$ is $K$-stable and thus admits a Kaehler-Einstein metric.  Somehow, the existence of certain op en cylinder s on $X$ d etects thi s.  As we shall see, these ope n cylinders corr espond precisel y to the existe nce of $\mathbb{C}_+$-a ctions on the af fine cone over $X$.  I'll spend some time attempti ng to demystify these ideas with plenty of explicit examples .< /span>