On asymptotics for the Mabuchi energy functional

Abstract

If $M$ is a projective manifold in $P^N$, then one can associate to each one parameter subgroup $H$ of $SL(N+1)$ the Mumford $\mu$ invariant. The manifold $M$ is Chow-Mumford stable if $\mu$ is positive for all $H$. Tian has defined the notion of K-stability, and has shown it to be intimately related to the existence of Kähler-Einstein metrics. The manifold $M$ is K-stable if ${\mu }^{\prime }$ is positive for all $H$, where ${\mu }^{\prime }$ is an invariant which is defined in terms of the Mabuchi K-energy. In this paper we derive an explicit formula for ${\mu }^{\prime }$ in the case where $M$ is a curve. The formula is similar to Mumford's formula for $\mu$, and is likewise expressed in terms of the vertices of the Newton diagram of a basis of holomorphic sections for the hyperplane line bundle.

14 pages