Stability and coercivity for toric polarizations

We introduce uniform K-stability and its relationship with the coercivity property of the K-energy functional, for general polarized manifolds. Since the automorphism groups are not necessarily finite, size of the norm measuring uniformity should be reduced with respect to the group action. About this point we explain that it is enough to take the reduced norm for a single sub-torus, actually the center, in the cscK problem. Our main theorem then describes the slope of the reduced J-functional along any torus-equivariant test configuration. In the toric case it is shown that the uniform stability is indeed equivalent to the coercivity of the K-energy. In the Fano manifolds case existence of the KE metric implies the uniform stability.