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 (in: Conference board of the mathematical sciences regional, conference series in mathematics, no. 7, American Mathematical Society, Providence, 1971), Licherowicz (Géométrie des groupes de transformation, Dunod, 1958), and Calabi (Extremal Kähler metrics, seminar on differential geometry, Princeton University Press, Princeton, 1982) in the Kähler–Einstein, cscK, and extremal Kähler cases. Combined with previous results of LeBrun (Commun Math Phys 344:621–653, 2016), Apostolov–Maschler (Conformally Kähler, Einstein–Maxwell geometry, arXiv:1512.06391v1) and Futaki–Ono (Volume minimization and conformally Kähler, Einstein–Maxwell geometry, arXiv:1706.07953), 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.