ScalingStacks

Proof. For any choice of $\theta_{i}$ the function $\phi(z_{1}e^{i\theta_{1}},\ldots z_{p}e^{i\theta_{n}},z_{p+1},\ldots,z_{n})$ is psh, since the $T^{p}$-action on $(\mathbb{C}^{*})^{p}$ is holomorphic. Thus the average function $\bar{\phi}$ is also psh as a function of $z_{1},\ldots z_{p}$. Any $T^{p}$-invariant psh function must be convex in the log coordinates, because for $x_{i}=\log|z_{i}|$,

$$\sqrt{-1}\partial\bar{\partial}\bar{\phi}=\frac{1}{4}\sum\frac{\partial^{2}\bar{\phi}}{\partial x_{i}\partial x_{j}}\sqrt{-1}d\log z_{i}\wedge d\overline{\log z_{j}}\geq 0.$$

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