ScalingStacks

Proof. (Thm. 5.1) There are two subcases: the interior of the top dimensional faces of $\partial\Delta_{\lambda}^{\vee}$, and the star of the vertices $\text{Star}(w)$. Since the arguments are almost the same we focus on the latter.

On the interior of $\text{Star}(w)$, we have local affine coordinates $x^{m_{1}},\ldots x^{m_{n}}$, related to the holomorphic $\mathbb{C}^{*}$-coordinates $z^{m_{1}},\ldots z^{m_{n}}$ by $x^{m_{i}}=s^{-1}\log|z^{m_{i}}|$. The star type region $U^{s,*}_{w}\subset X_{s}$ can be viewed as a subset of $(\mathbb{C}^{*})^{n}$, so we use the rescaled map $s^{-1}\text{Log}:(\mathbb{C}^{*})^{n}_{z^{m_{i}}}\to\mathbb{R}^{n}_{x^{m_{i}}}$ to pullback the function $u_{\infty,m}$ on $\text{Star}(w)$. On the other hand, $X_{s}\cap(\mathbb{C}^{*})^{n+1}$ maps into $N_{\mathbb{R}}$ via $\text{Log}_{s}$, so we can also pullback $u_{\infty,m}$ via $\text{Log}_{s}$. These two pullbacks differ by at most $Cs^{-1}$ using Cor. 4.15. We also write $\phi=\varphi_{CY,s,m}$.

Take a local test function $f\in C^{2}_{c}$ supported in the interior of $\text{Star}(w)$, then $f$ is identified as a local function on $U^{s,*}_{w}\subset X_{s}$ via $s^{-1}\text{Log}$. By the Chern-Levine type estimate above,

$$\begin{split}&s^{n}\int f\{(\sqrt{-1}\partial\bar{\partial}\phi)^{n}-(\sqrt{-1}\partial\bar{\partial}(u_{\infty,m}\circ s^{-1}\text{Log}))^{n}\}\\ &\leq C\left\lVert\phi-u_{\infty,m}\circ s^{-1}\text{Log}\right\rVert_{L^{\infty}}\left\lVert f\right\rVert_{C^{2}}\to 0,\end{split}$$

as $s\to+\infty$. By the Calabi-Yau condition (20) and Prop. 3.14,

$$s^{n}(\sqrt{-1}\partial\bar{\partial}\phi)^{n}=a_{s}d\mu_{s}=a_{\infty}d\mu_{s}(1+o(1)),\quad s\to+\infty.$$

Pushing forward via $s^{-1}\text{Log}$, and applying Lemma 5.2,

$$\pi^{n}n!\int fMA(u_{\infty})=\lim_{s\to\infty}\int fa_{s}(s^{-1}\text{Log})_{*}d\mu_{s}=a_{\infty}\int fd\mu_{\infty}.$$

Since this holds for every $f\in C^{2}_{c}$, on the interior of this top dimensional face we obtain the measure equality (31). ∎