ScalingStacks

The holomorphic volume form (9) is

$$\begin{split}\Omega_{s}=&\pm\frac{d\log z^{\mathfrak{m}_{1}}\wedge\ldots d\log z^{\mathfrak{m}_{l}}\wedge d\log z^{m_{0}}\wedge\ldots d\log z^{m_{n-l}}}{dF_{s}}\\ =&\pm\frac{dz^{\mathfrak{m}_{1}}\wedge\ldots dz^{\mathfrak{m}_{l}}}{z^{-m_{0}^{\prime}}}\bigwedge\frac{d\log z^{m_{0}}\wedge\ldots d\log z^{m_{n-l}}}{dF_{s}}\\ \approx&\pm dz^{\mathfrak{m}_{1}}\wedge\ldots dz^{\mathfrak{m}_{l}}\bigwedge d\log z^{m_{1}}\wedge\ldots d\log z^{m_{n-l}}\bigwedge\frac{d\log z^{m_{0}}}{a_{m_{1}^{\prime}}e^{s\lambda(m_{1}^{\prime})}dz^{m_{1}^{\prime}-m_{0}^{\prime}}}\\ =&\frac{dz^{\mathfrak{m}_{1}}\wedge\ldots dz^{\mathfrak{m}_{l}}}{a_{m_{1}^{\prime}}e^{s\lambda(m_{1}^{\prime})}z^{m_{1}^{\prime}-m_{0}^{\prime}}\mathfrak{d}}\bigwedge d\log z^{m_{1}}\wedge\ldots d\log z^{m_{n-l}},\end{split}$$

(13)

up to choosing appropriate ordering of the coordinates. Here $\mathfrak{d}$ is the divisibility of $m_{1}^{\prime}-m_{0}^{\prime}$ inside the group

$$\text{span}_{\mathbb{Q}}\{m_{1}^{\prime}-m_{0}^{\prime},\ldots,m_{\dim\sigma}^{\prime}-m_{0}^{\prime}\}\cap M/\text{span}_{\mathbb{Z}}\{m_{1},\ldots,m_{\dim\sigma-1}\}\simeq\mathbb{Z}.$$

Notice $a_{m_{1}^{\prime}}e^{s\lambda(m_{1}^{\prime})}z^{m_{1}^{\prime}-m_{0}^{\prime}}$ is uniformly equivalent to $a_{m_{0}^{\prime}}e^{s\lambda(m_{0}^{\prime})}$ in this region.