Equivariant K-stability and Valuative Criteria

It is conjectured that in order to test K-polystability of a Fano variety, it is enough to consider only equivariant test configurations with respect to a finite or connected reductive group action. This has been proved for Fano manifolds and in singular cases, only for torus actions. In this talk, I will talk about a valuative criterion of equivariant K-stability with respect to an arbitrary group action. This generalizes parallel results for usual K-stability.