Volume minimization and Conformally K\"ahler, Einstein-Maxwell geometry

Let $M$ be a compact complex manifold admitting a K\"ahler structure. A conformally K\"ahler, Einstein-Maxwell metric (cKEM metric for short) is a Hermitian metric $\tilde{g}$ on $M$ with constant scalar curvature such that there is a positive smooth function $f$ with $g = f^2 \tilde{g}$ being a K\"ahler metric and $f$ being a Killing Hamiltonian potential with respect to $g$. Fixing a K\"ahler class, we characterize such Killing vector fields whose Hamiltonian function $f$ with respect to some K\"ahler metric $g$ in the fixed K\"ahler class gives a cKEM metric $\tilde{g} = f^{-2}g$. The characterization is described in terms of critical points of certain volume functional. The conceptual idea is similar to the cases of K\"ahler-Ricci solitons and Sasaki-Einstein metrics in that the derivative of the volume functional gives rise to a natural obstruction to the existence of cKEM metrics. However, unlike the K\"ahler-Ricci soliton case and Sasaki-Einstein case, the functional is neither convex nor proper in general, and often has more than one critical points. The last observation matches well with the ambitoric examples studied earlier by LeBrun and Apostolov-Maschler.