Conformally Einstein-Maxwell Kahler metrics and structure of the automorphism group

Let (M, g) be a compact Kahler manifold and f a positive smooth function such that its Hamiltonian vector field for the Kahler form is a holomorphic Killing vector field. We say that the pair (g, f) is conformally Einstein-Maxwell Kahler metric if the conformal metric has constant scalar curvature. In this paper we prove a reductiveness result of the reduced Lie algebra of holomorphic vector fields for conformally Einstein-Maxwell Kahler manifolds, extending the Lichnerowicz-Matsushima Theorem for constant scalar curvature Kahler manifolds. More generally we consider extensions of Calabi functional and extremal Kahler metrics, and prove an extension of Calabi's theorem on the structure of the Lie algebra of holomorphic vector fields for extremal Kahler manifolds. The proof uses a Hessian formula for the Calabi functional under the set up of Donaldson-Fujiki picture.