Volume minimization and Conformally Kähler, Einstein-Maxwell geometry

Abstract

Let $M$ be a compact complex manifold admitting a Kähler structure. A conformally Kähler, 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 Kähler metric and $f$ being a Killing Hamiltonian potential with respect to $g$. Fixing a Kähler class, we characterize such Killing vector fields whose Hamiltonian function $f$ with respect to some Kähler metric $g$ in the fixed Kähler 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 Kähler-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 Kähler-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.