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let $(M,\omega)$ be a Kahler-Einstein toric manifold of complex dimension $m$. By toric manifold i mean a manifold that has an open dense subset $X$ biholomorphic to an algebraic torus $\mathbb{T}^{m}:=( \mathbb{C}^{*})^{m}$ and there is a holomorphic action \begin{equation} \alpha:\mathbb{T}^{m}\times M\rightarrow M \end{equation} that on $X$ restricts to the standard action of $\mathbb{T}^{m}$ on itself. My question is the following: is the Kahler Einstein metric $\omega$ automatically invariant under the action of $\mathbb{T}^{m}$? If it is a known result can someone tell me a reference?

Thank you in advance.

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  • $\begingroup$ Well, the Fubini-Study metric on $\mathbb P^1$ is already not invariant under the torus action. $\endgroup$
    – Henri
    Commented Jan 29, 2014 at 14:33
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    $\begingroup$ Your notation is causing some confusion because you are confusing the algebraic torus with the compact subgroup $K$ generated by the $S^1$-subgroups of the $\mathbb{C}^\ast$-factors, and $K$ is a topological torus. Generally, one asks that the toric metric be invariant under $K$, not the whole (noncompact) algebraic torus, which is much more reasonable. $\endgroup$
    – Robert Bryant
    Commented Jan 29, 2014 at 15:35
  • $\begingroup$ @Robert: If i consider the action of the compact subgroup $K$ is $\omega$ K-invariant? Are there obstructions? $\endgroup$
    – Italo
    Commented Jan 29, 2014 at 15:44

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I would put this in the comments if I could. I think you will find the answer to your question here (I cannot access this paper):

Y. Matsushima: Sur la structure de groupe d'homeomorphismes analytiques d'une certaine variete kaehlerienne, Nagoya Math. J. 11 (1957), 145-150.

as quoted in Theorem 2.4 in

T. Mabuchi: Einstein-Kähler forms, Futaki invariants and convex geometry on toric Fano varieties

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