Seminar Algebra and Geometry: Giulio Codogni (Roma Tre)

Torus equivariant K-stability

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. This is a joint work with J. Stoppa.

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