It is conjectured that to test the K-polystability of a polarised variety it is enough to consider test-configurations which are equivariant with respect to a torus in the automorphism group. We prove partial results towards this conjecture. We also show that it would give a new proof of the K-polystability of constant scalar curvature polarised manifolds.
Codogni, G., Stoppa, J. (2019). Torus equivariant K-stability. In Springer INdAM Series (pp. 15-35). Springer International Publishing [10.1007/978-3-030-13158-6_2].