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.
Ying-Ji HONG. "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