Lemma 5.2 (00SL) · ScalingStacks

00SL

Lemma 5.2. Let $u$ be a bounded convex function on the square $\{|x_{i}|<1\}\subset\mathbb{R}^{n}$ . Via the rescaled log map $s^{-1}\text{Log}:(\mathbb{C}^{*})^{n}\to\mathbb{R}^{n}$ , the function $u$ pulls back to a psh function on $\{|\log|z_{i}||<s\}$ . Then the real MA measure of $u$ is related to the pushforward of the complex MA measure of $u\circ s^{-1}\text{Log}$ by

MA(u)=\frac{s^{n}}{\pi^{n}n!}(s^{-1}\text{Log})_{*}(\sqrt{-1}\partial\bar{\partial}(u\circ s^{-1}\text{Log}))^{n}.

00SM

Proof. If $u$ is smooth, then

MA(u)(K)=\int_{K}\det(D^{2}u)dx_{1}\ldots dx_{n}=\frac{s^{n}}{\pi^{n}n!}\int_{(s^{-1}\text{Log})^{-1}(K)}(\sqrt{-1}\partial\bar{\partial}(u\circ s^{-1}\text{Log}))^{n}.

Since $u\in C^{0}$ , and both the real and complex MA operators are weakly continuous with respect to $C^{0}$ -limits, this equality passes to general $u$ . ∎