CM Stability And The Generalized Futaki Invariant II

Abstract

The Mabuchi K-energy map is exhibited as a singular metric on the refined CM polarization of any equivariant family $\mathbf{X}\stackrel{p}{\to }S$. 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 added