| The first part of the talk will be an introduction to K-stability. K-stability, conjecturally, characterizes the varieties which admit a constant scalar curvature Kahler metric. A goal of this theory is to generalize the uniformization of Riemann surfaces to higher dimensional varieties. In the second part of the talk I will explain a joint work with R. Dervan. We study the K-stability of a polarised variety (X,L) with non-reductive automorphism group. We associate a canonical filtration of the co-ordinate ring to each variety of this kind, which destabilises the variety in several examples which we compute. We conjecture this holds in general. This is an algebro-geometric analogue of Matsushima's theorem regarding the existence of constant scalar curvature Kahler metrics. As an application, we give an example of an orbifold del Pezzo surface without a Kahler-Einstein metric. |