Automorphisms and deformations of conformally K\"ahler, Einstein-Maxwell metrics

We obtain a structure theorem for the group of holomorphic automorphisms of a conformally K\"ahler, Einstein-Maxwell metric, extending the classical results of Matsushima, Licherowicz and Calabi in the K\"ahler-Einstein, cscK, and extremal K\"ahler cases. Combined with previous results of LeBrun, Apostolov-Maschler and Futaki-Ono, this completes the classification of the conformally K\"ahler, Einstein--Maxwell metrics on $\mathbb{{CP}}^1 \times \mathbb{{CP}}^1$. We also use our result in order to introduce a (relative) Mabuchi energy in the more general context of $(K, q, a)$-extremal K\"ahler metrics in a given K\"ahler class, and show that the existence of $(K, q, a)$-extremal K\"ahler metrics is stable under small deformation of the K\"ahler class, the Killing vector field $K$ and the normalization constant $a$.