On asymptotics for the Mabuchi energy functional

Details On asymptotics for the Mabuchi energy functional On asymptotics for the Mabuchi energy functional

2003-12-31

arxiv.org/abs/math/0312528v1

Mathematics

Differential Geometry

14 pages

Scientific paper

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 K\"ahler-Einstein metrics. The manifold $M$ is K-stable if $\mu'$ is positive for all $H$, where $\mu'$ is an invariant which is defined in terms of the Mabuchi K-energy. In this paper we derive an explicit formula for $\mu'$ 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.

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