| We prove (using algebro-geometric methods) two results that allow to test the positivity of the Donaldson-Futaki weights of arbitrary polarised varieties via test-configurations which are equivariant with respect to a maximal torus in the automorphism group. It follows in particular that there is a purely algebro-geometric proof of the K-stability of projective spaces (or more generally of smooth toric Fanos with vanishing Futaki character, as well as of the examples of non-toric Kahler-Einstein Fano threefolds due to Ilten and Suss) and that K-stability for toric polarised manifolds can be tested via toric test-configurations. A further application is a proof of the K-stability of constant scalar curvature polarised manifolds with continuous automorphisms. Our approach is based on the method of filtrations introduced by Wytt Nystrom and Szekelyhidi and indeed many of our relts also extend to the class of (not necessarily finitely generated) polynomial filtrations. This is a joint work with J. Stoppa. |