ScalingStacks

The Calabi-Yau measure (2) is studied thoroughly in [3], but it is illustrative to recall it explicitly on a semistable snc model $\mathcal{X}$. The discussion is local on the base, and we will follow the notations of section 2, e.g. the components of the central fibre are denoted as $E_{i}$ for $i\in I$.

The canonical divisor $K_{\mathcal{X}}=\sum_{i}a_{i}E_{i}$ is supported on the central fibre, since $K_{X}$ is trivialised. Multiplying $\Omega$ by a power of $t$, which does not change $d\mu_{t}$, we may assume $\min a_{i}=0$. In the local coordinates around $E_{J}$ away from the deeper strata,

$$\Omega=f_{J}\prod_{0}^{p}z_{i}^{a_{i}}dz_{i}\wedge\prod_{p+1}^{n}dz_{j}$$

for some nowhere vanishing local holomorphic function $f_{J}$. Since $t=z_{0}\ldots z_{p}$,

$$\Omega_{t}=f_{J}z_{0}^{a_{0}}\ldots z_{p}^{a_{p}}\prod_{1}^{p}d\log z_{i}\wedge\prod_{p+1}^{n}dz_{j},$$

hence

$$\sqrt{-1}^{n^{2}}\Omega_{t}\wedge\overline{\Omega}_{t}=|f_{J}|^{2}|z_{0}|^{2a_{0}}\ldots|z_{p}|^{2a_{p}}\prod_{1}^{p}\sqrt{-1}d\log z_{i}\wedge d\overline{\log z_{i}}\wedge\prod_{p+1}^{n}\sqrt{-1}dz_{j}\wedge d\overline{z}_{j}.$$

The total measure $\int_{X_{t}}\sqrt{-1}^{n^{2}}\Omega_{t}\wedge\overline{\Omega}_{t}$ is of the order $O(|\log|t||^{m})$ where

$$m=\max\{|J|-1:E_{J}\neq\emptyset,a_{i}=0\text{ for }i\in J\}.$$

Thus $d\mu_{t}$ satisfies a uniform upper bound of class $(a_{i})$ (cf. (4)).