A K-energy functional for complexified Kähler classes

Abstract

Given a complexified Kähler class on a compact Kähler manifold, we introduce a generalisation of the K-energy functional, defined on the space of complexified Kähler potentials, whose critical points are solutions of the scalar curvature equation with B-field introduced by Schlitzer and Stoppa. We prove that this extended K-energy is convex along geodesics in the space of almost calibrated potentials. As an application, we show that, at least in some notable cases, solutions of the scalar curvature equation with B-field are unique in their class, confirming the expectation that the scalar curvature equation with B-field identifies canonical representatives of complexified Kähler classes.