Volume minimization and conformally Kaehler, Einstein-Maxwell geometry

摘要

Let M be a compact complex manifold admitting a Kaehler structure. A conformally Kaehler, Einstein-Maxwell metric (cKEM metric for short) is a Hermitian metric g on M with constant scalar curvature such that there is a positive smooth function f with g = f~2g being a Kaehler metric and f being a Killing Hamiltonian potential with respect to g. Fixing a Kaehler class, we characterize such Killing vector fields whose Hamiltonian function f with respect to some Kaehler metric g in the fixed Kaehler class gives a cKEM metric 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 Kaehler-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 Kaehler-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.