We prove that if a closed polarized complex manifold admits a conformally K\"ahler, Einstein--Maxwell metric, or more generally, a K\"ahler metric of constant $(\xi, a, p)$-scalar curvature, then it minimizes the $(\xi,a,p)$-Mabuchi functional. Our method of proof extends the approach introduced by Donaldson, via finite dimensional approximations and generalized balanced metrics. As an application of our result and the recent construction of Koca--T{\o}nnesen-Friedman, we describe the K\"ahler classes on a geometrically ruled complex surface of genus greater than 2, which admit conformally K\"ahler, Einstein-Maxwell metrics. 查看全文>>