Abstract
Suppose that M is a compact Fano manifold. That is, M is a compact Kahler manifold with positive first Chern class. One of the most important problems in Kahler geometry is the existence of Kahler metrics of constant scalar curvature. It is believed that the problem is related to certain notion of stability in the sense of Geometric Invariant Theory. In Tian [17] and Donaldson [4], the notion of K stability was introduced. In the first three sections of this paper, we use the notations in [17] to derive our theorems. In the last section, we discuss the definition of [4] and some observations motivated by that paper. Let M be a Fano manifold that is embedded in CPn by the k-th power of the anticanonical line bundle, where k is a positive integer. Let σ(t) be a one parameter family of automorphisms of CPn. We write
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