2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/159358The Mabuchi K-energy map is exhibited as a singular metric on the refined CM polarization of any equivariant family $\mathbf{X}\overset{p}{\to} S$. Consequently we show that the generalized Futaki invariant is the leading term in the asymptotics of the reduced K-energy of the generic fiber of the map $p$. Properness of the K-energy implies that the generalized Futaki invariant is strictly negative.15, fully rewritten, references addedDifferential GeometryAlgebraic Geometry53C55 (Primary), 14D06 (Secondary)text