Conformally Einstein-Maxwell K\"ahler metrics and structure of the automorphism group

Let $(M,g)$ be a compact K\"ahler manifold and $f$ a positive smooth function such that its Hamiltonian vector field $K = J\mathrm{grad}_g f$ for the K\"ahler form $\omega_g$ is a holomorphic Killing vector field. We say that the pair $(g,f)$ is conformally Einstein-Maxwell K\"ahler metric if the conformal metric $\tilde g = f^{-2}g$ 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 K\"ahler manifolds, extending the Lichnerowicz-Matsushima Theorem for constant scalar curvature K\"ahler manifolds. More generally we consider extensions of Calabi functional and extremal K\"ahler metrics, and prove an extension of Calabi's theorem on the structure of the Lie algebra of holomorphic vector fields for extremal K\"ahler manifolds. The proof uses a Hessian formula for the Calabi functional under the set up of Donaldson-Fujiki picture.