In this article we discuss how the existence of Kaehler metrics with constant scalar curvature on the projectivization of a holomorphic vector bundle over a Kaehler manifold is related to a moment map condition for the action of the automorphism group of on the moduli of vector bundles.
"Stability and existence of critical Kaehler metrics on ruled manifolds." J. Math. Soc. Japan 60 (1) 265 - 290, January, 2008. https://doi.org/10.2969/jmsj/06010265