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On asymptotics for the Mabuchi energy functional

D. H. Phong·Jacob Sturm
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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 $μ$ invariant. The manifold $M$ is Chow-Mumford stable if $μ$ 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 $μ'$ is positive for all $H$, where $μ'$ is an invariant which is defined in terms of the Mabuchi K-energy. In this paper we derive an explicit formula for $μ'$ in the case where $M$ is a curve. The formula is similar to Mumford's formula for $μ$, 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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D. H. Phong, Jacob Sturm. 2003-12-31. On asymptotics for the Mabuchi energy functional. https://arxiv.org/abs/math/0312528

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