The yau-tian-donaldson conjecture for general polarizations, I

Abstract

In this paper, some C0 boundedness property (BP) is introduced for balanced metrics on a polarized algebraic manifold (X, L). Then by assuming that (X, L) is strongly K-stable in the sense of [8], we shall show that the balanced metrics have (BP). In a subsequent paper [10], this property (BP) plays a very important role in the study of the Yau-Tian-Donaldson conjecture for general polarizations.