There is a quantitative stratification on any smooth fibre $X_{t}$ induced by the intersection pattern of $E_{i}$: for $J\subset I$ such that $E_{J}=\cap_{i\in J}E_{i}\neq\emptyset$, the corresponding statum is

namely a small ‘$\epsilon$-tubular neighbourhood’ of $E_{J}$ minus the deeper strata. For $J=\{i\}$ we write $E_{i}^{0}=E_{\{i\}}^{0}$. Here the disc $\mathbb{D}_{t}$ and the small parameter $\epsilon\ll 1$ can be shrinked for convenience; the essential thing is that all parameters should be independent of the coordinate $t$.

It is useful to introduce local coordinates $\{z_{i}\}_{0}^{n}$ around $E_{J}\subset\mathcal{X}$, such that $z_{0},\ldots,z_{p}$ with $p=|J|-1$ are the local defining equations of $E_{j}$ for $j\in J$, and locally the fibration map is $t=z_{0}\ldots z_{p}$. Then up to uniform equivalence, locally

The rest of this section is devoted to the construction of good test functions. Given any of these divisors $E_{0}$, we can find a nonnegative function $h=h_{E_{0}}$ on $\mathcal{X}$, such that

• •

  In the local charts near $E_{0}$ with $z_{0}$ being the defining function for $E_{0}$,

  $$h=|z_{0}|^{2}\tilde{h}(z_{0},\ldots z_{n})$$

  for some positive smooth function $\tilde{h}$;

• •

  Away from $E_{0}$ the function $h$ is comparable to 1.

We observe

• •

  The form $\partial\bar{\partial}\log h=\partial\bar{\partial}\log\tilde{h}$ extends smoothly;

• •

  For $|t|^{2}\ll h\lesssim\delta\ll 1$ inside $X_{t}$, so that $|z_{0}|\gg|t|$, by a local calculation near $E_{J}$ with $0\in J$,

  $$\begin{split}&\sqrt{-1}\partial\log h\wedge\bar{\partial}\log h\wedge\omega_{\mathcal{X}}|_{X_{t}}^{n-1}\geq\frac{\sqrt{-1}}{2|z_{0}|^{2}}dz_{0}\wedge d\bar{z}_{0}\wedge\omega_{\mathcal{X}}|_{X_{t}}^{n-1}\\ &\gtrsim\min\{\frac{1}{|z_{0}|^{2}},\max_{1\leq i\leq p}|z_{i}|^{-2}\}\omega_{\mathcal{X}}|_{X_{t}}^{n}\\ &\gtrsim\min\{\frac{1}{h},h^{1/p}|t|^{-2/p}\}\omega_{\mathcal{X}}|_{X_{t}}^{n}\\ &\gtrsim\min\{\frac{1}{h},h^{1/n}|t|^{-2/n}\}\omega_{\mathcal{X}}|_{X_{t}}^{n}.\end{split}$$

  Here in the first line we need to fix $\delta\ll 1$ so that the effect of $\partial\log h$ is dominated by $d\log z_{0}$. The second line uses that for $1\leq k\leq p$, the volume forms on $X_{t}$

  $$\frac{1}{|z_{0}|^{2}}dz_{0}\wedge d\bar{z}_{0}\wedge\prod_{j\neq k,1\leq j\leq n}\sqrt{-1}dz_{j}\wedge d\bar{z}_{j}\sim\frac{1}{|z_{k}|^{2}}\prod_{1\leq j\leq n}\sqrt{-1}dz_{j}\wedge d\bar{z}_{j},$$

  and the third line uses $|t|=|z_{0}\ldots z_{p}|\sim h^{1/2}|z_{1}\ldots z_{p}|$.

• •

  On $X_{t}$ the function $h\gtrsim|t|^{2}$. The region $\{|t|^{2}\sim h\}\subset X_{t}$ can be identified as $E_{0}^{0}$, namely the vicinity of $E_{0}$ away from deeper strata. Here

  $$\begin{split}&\sqrt{-1}\partial\log h\wedge\bar{\partial}\log h\wedge\omega_{\mathcal{X}}|_{X_{t}}^{n-1}\geq 0,\\ &\sqrt{-1}\partial\bar{\partial}\log h\wedge\omega_{\mathcal{X}}|_{X_{t}}^{n-1}\gtrsim-\omega_{\mathcal{X}}|_{X_{t}}^{n}.\end{split}$$