Uniform stability and coercivity for equivariant polarizations

This is a publication of the author's old preprint which extends the notion of uniform K-stability and the related coercivity property of the K-energy functional to general polarized manifolds which possibly have continuous automorphism group. The result is utilized in subsequent research and there were considerable developments on the subject, however, this paper is presented as it was. Our main theorem describes the slope of the reduced J-functional along any equivariant test configuration. In the toric case it follows that the uniform stability is equivalent to the coercivity of the K-energy. For Fano manifolds existence of the KE metric implies the uniform stability.