Title: 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 $a$.