Automorphisms and deformations of conformally Kähler, Einstein-Maxwell metrics

Abstract

We obtain a structure theorem for the group of holomorphic automorphisms of a conformally Kähler, Einstein-Maxwell metric, extending the classical results of Matsushima, Licherowicz and Calabi in the Kähler-Einstein, cscK, and extremal Kähler cases. Combined with previous results of LeBrun, Apostolov-Maschler and Futaki-Ono, this completes the classification of the conformally Kähler, 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 Kähler metrics in a given Kähler class, and show that the existence of $(K,q,a)$-extremal Kähler metrics is stable under small deformation of the Kähler class, the Killing vector field $K$ and the normalization constant $a$.