A Variational Characterization of Positive Scalar Curvature K\"ahler metrics

Abstract

We introduce the prescribed scalar curvature measure equation on a compact K\"ahler manifold. For a K\"ahler class of positive total scalar curvature, we prove that the following are equivalent: the existence of a positive scalar curvature K\"ahler metric, solvability of this equation for every admissible measure, $d_1$-coercivity of the associated functionals, and uniform geodesic stability along finite-energy $d_1$-geodesic rays. As a consequence, in each fixed K\"ahler class, the space of positive scalar curvature K\"ahler metrics is either empty or contractible. We further prove that every K\"ahler class on a positive-dimensional compact smooth toric K\"ahler manifold contains a torus-invariant metric of positive scalar curvature. Therefore, for every K\"ahler class, the prescribed scalar curvature measure equation admits a smooth solution for every admissible measure, unique modulo constants.