ScalingStacks

Proposition 3.1. We can find a metric $\omega_{t}^{\prime}$ on $X_{t}$ in the class $\frac{1}{|\log|t||}c_{1}(\mathfrak{L})|_{X_{t}}$, a Lipschitz function $\rho_{t}$ on $X_{t}$, and an open neighborhood $U$ of $E_{J}$ in $\mathfrak{X}$ as above, with the following properties:

• (a)

  The function $\rho_{t}$ is supported on the closure of $B_{t}=\{\rho_{t}<0\}\subset U\cap X_{t}$. On $B_{t}$ the function $\rho_{t}$ is comparable to a quadratic function in the logarithmic variables $x_{j}=\frac{\log|z_{j}|}{|\log|t||}$, for $j=1,2,\ldots m$, in adapted coordinate charts, with $\min\rho_{t}=-1$ and $\max\rho_{t}=0$.

• (b)

  On $B_{t}$ in adapted coordinate charts we have

  (3.2)

  $$\omega_{t}^{\prime}\geqslant C^{-1}\frac{i}{|\log|t||^{2}}\sum_{j=1}^{m}\frac{dz_{j}}{z_{j}}\wedge\frac{d\overline{z_{j}}}{\overline{z_{j}}},$$

  and

  (3.3)

  $$|d\rho_{t}|^{2}_{\omega^{\prime}_{t}}\leqslant C,$$

  for a fixed constant $C$ independent of $t$.