We prove that a metric of constant scalar curvature on a polarised Kähler manifold is the limit of metrics induced from a specific sequence of projective embeddings; satisfying a condition introduced by H. Luo. This gives, as a Corollary, the uniqueness of constant scalar curvature Kähler metrics in a given rational cohomology class. The proof uses results in the literature on the asymptotics of the Bergman kernel. The arguments are presented in a general framework involving moment maps for two different group actions.
"Scalar Curvature and Projective Embeddings, I." J. Differential Geom. 59 (3) 479 - 522, November, 2001. https://doi.org/10.4310/jdg/1090349449