Abstract: We will introduce a notion of a Kahler metric with constant weighted scalar curvature on a compact Kaehler manifold X, depending on a fixed real torus T in the reduced group of automorphisms of X, and two smooth (weight) functions dened on the momentum image of X. We will also dene a notion of weighted Mabuchi energy adapted to our setting, and of a weighted Futaki invariant of a T-compatible smooth Kaehler test conguration associated to (X; T). After that, using the geometric quantization scheme of Donaldson, we will show that if a projective manifold admits in the corresponding Hodge Kaehler class a Kaehler metric with constant weighted scalar curvature, then this metric minimizes the weighted Mabuchi energy, which implies a suitable notion of weighted K-semistability. As an application, we describe the Kaehler classes on a geometrically ruled complex surface of genus greater than 2,which admits conformally Kaehler Einstein-Maxwell metrics.
