ScalingStacks

Let us now recall the asymptotic behavior of the integrals $\int_{X_{t}}i^{n^{2}}\Omega_{t}\wedge\overline{\Omega_{t}}$, largely following [3]. For any $J\subset I$ we denote by $E_{J}=\bigcap_{j\in J}E_{j}$. As in [13, 14], we fix a Kähler metric on $\mathfrak{X}$ and for $\varepsilon>0$ small and $J\subset I$ with $E_{J}\neq\emptyset$ we define

$$E^{0}_{J}=\{x\in X_{t}\ |\ d(x,E_{J})<\varepsilon\}\backslash\{x\in X_{t}\ |\ d(x,E_{J^{\prime}})<\varepsilon\text{ for some }J^{\prime}\supsetneq J\}.$$

For any given $x\in E_{J}\subset\mathfrak{X}$ let $p=|J|-1$ and pick local coordinates $z_{0},\dots,z_{n}$ on $x\in V\subset\mathfrak{X}$, defined in the unit polydisc, such that $z_{0},\dots,z_{p}$ are defining equations for $E_{j},j\in J$, so that in these coordinates we have $t=z_{0}\cdots z_{p}$. We shall call these adapted coordinates. We can then write

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

where $f_{J}$ is a local non-vanishing holomorphic function. Since $dt\wedge\Omega_{t}=\Omega$ along $X_{t}$, on $E^{0}_{J}$ we get

$$\Omega_{t}=f_{J}z_{0}^{a_{0}}\cdots z_{p}^{a_{p}}\prod_{j=1}^{p}\frac{dz_{j}}{z_{j}}\wedge\prod_{k=p+1}^{n}dz_{k},$$

$$i^{n^{2}}\Omega_{t}\wedge\overline{\Omega_{t}}=|f_{J}|^{2}|z_{0}|^{2a_{0}}\cdots|z_{p}|^{2a_{p}}\prod_{j=1}^{p}i\frac{dz_{j}}{z_{j}}\wedge\frac{d\overline{z_{j}}}{\overline{z_{j}}}\wedge\prod_{k=p+1}^{n}idz_{k}\wedge d\overline{z_{k}},$$

from which using polar coordinates $z_{j}=\exp(x_{j}\log|t|+i\theta_{j}),j\in J,$ one can easily see as in [3] that

$$\int_{E^{0}_{J}}i^{n^{2}}\Omega_{t}\wedge\overline{\Omega_{t}}\sim|\log|t||^{m_{J}},$$

where

$$m_{J}=|\{j\in J\ |\ a_{j}=0\}|-1,$$

while

$$\int_{X_{t}}i^{n^{2}}\Omega_{t}\wedge\overline{\Omega_{t}}\sim|\log|t||^{m},$$

where

(2.2)

$$m=\max\{|J|-1\ |\ E_{J}\neq\emptyset,a_{j}=0\text{ for all }j\in J\}=\dim\mathrm{Sk}(X).$$

The local logarithmic variables $x_{j}=\frac{\log|z_{j}|}{\log|t|}$ vary in the standard simplex

$$\Delta_{J}=\left\{0\leqslant x_{j}\leqslant 1\ |\ \sum_{j=0}^{p}x_{j}=1\right\},$$

and in this way one obtains a map $\mathrm{Log}_{t}:V\backslash\bigcup_{j}E_{j}\to\Delta_{J}$, see [3].