Lemma 4.3 (00RE) · ScalingStacks

00RE

Lemma 4.3. Let $\Phi$ be any psh function on the open subset of $\{1<|\zeta_{i}|<\Lambda,i=1,\ldots n\}\subset(\mathbb{C}^{*})^{n}$ . Then the $T^{n}$ -invariant function

\bar{\Phi}(\log|\zeta_{1}|,\ldots,\log|\zeta_{n}|)=\frac{1}{(2\pi)^{n}}\int_{T^{n}}\Phi(|\zeta_{1}|e^{i\theta_{1}},\ldots|\zeta_{n}|e^{i\theta_{n}})d\theta_{1}\ldots d\theta_{n}

is a convex function in the variables $x_{1}=\log|\zeta_{1}|,\ldots,x_{n}=\log|\zeta_{n}|$ .

00RF

Proof. Since the $T^{n}$ -action on $(\mathbb{C}^{*})^{n}$ is holomorphic, $\Phi(\zeta_{1}e^{i\theta_{1}},\ldots\zeta_{n}e^{i\theta_{n}})$ is psh in $\zeta$ for any choice of $\theta_{i}$ , so the average function $\bar{\Phi}$ is also psh. Any $T^{n}$ -invariant psh function must be convex in the log coordinates, because of the formula

\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\zeta_{i}\wedge d\overline{\log\zeta_{j}}\geq 0.

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