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One definition of Toric Kahler manifolds that I know is the following - $(M,J,\omega)$ is a Kahler manifold of dimension n with an $\textit{effective}$ $\textbf{T}^n$ action which preserves the symplectic form and the complex structure, where $\textbf{T}^n$ is the real torus of dimension n. Then

The action of $\textbf{T}^n$ extends to an action of $(\textbf{C}^*)^n$ on $(M,J,\omega)$. Is it automatically true that this action has a free, open, dense orbit sitting inside $M$?

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This is true for any toric manifold (or toric variety): en.wikipedia.org/wiki/Toric_variety . In your definition you should add that $dim M=2n$. –  Dmitri Jun 6 '11 at 20:52

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