Given any point $x\in E_{J}$ we can find $k\gg 1$ and sections $s_{0},\dots,s_{N}\in H^{0}(\mathfrak{X},\mathfrak{L}^{k})$ so that in some adapted coordinate chart $V_{x}$ near $x$ we have that none of the sections $s_{0},s_{m+1},\dots,s_{N}$ vanishes, while $s_{j}=0$ is a defining equation for $E_{j}$, $1\leqslant j\leqslant m$, and so $s_{j}/s_{0}$ is comparable to $z_{j}$ for $1\leqslant j\leqslant m$.

We construct a Kähler metric $\omega_{x,t}^{\prime}$ on $X_{t}$ by pulling back a suitable toric metric on $\mathbb{P}^{N}$ , which on the complement of the zeros of all the $s_{i}$’s is given by

$$\omega^{\prime}_{x,t}=\frac{1}{k}i\partial\bar{\partial}u\left(\frac{\log|s_{1}/s_{0}|}{\log|t|},\dots,\frac{\log|s_{N}/s_{0}|}{\log|t|}\right),$$

where $u(x_{1},\dots,x_{N})$ is a smooth convex function in $\mathbb{R}^{N}$ which is asymptotic to $v(x_{1},\dots,x_{N})=\max(0,x_{1},\dots,x_{N})$ at infinity, and with $D^{2}u\geqslant C^{-1}\mathrm{Id}$ on a ball of radius comparable to 1 containing the image of $V_{x}\cap X_{t}$ in the logarithmic coordinates. For example, an explicit such $u$ can be produced as the convolution of $v$ with a smooth mollifier. By construction, $\omega^{\prime}_{x,t}$ lies in the class $\frac{1}{|\log|t||}c_{1}(\mathfrak{L})|_{X_{t}}$, and it satisfies (3.2) on $V_{x}\cap X_{t}$.

We then choose finitely many $x^{(1)},\dots,x^{(M)}\in E_{J}$ such that the corresponding $V_{x^{(1)}},\dots,V_{x^{(M)}}$ cover $E_{J}$, let $U$ be their union, and define

$$\omega^{\prime}_{t}=\frac{1}{M}\sum_{k=1}^{M}\omega^{\prime}_{x^{(k)},t}.$$

This is our desired Kähler metric on $X_{t}$ in $\frac{1}{|\log|t||}c_{1}(\mathfrak{L})|_{X_{t}}$ which satisfies (3.2) in adapted coordinate charts on $U\cap X_{t}$.

Next, we consider the function

$$\hat{\rho}=A\sum_{j=1}^{m}\left(\frac{\tilde{x}_{j}}{2}-b_{j}\right)^{2}-1=A\sum_{j=1}^{m}\left(\frac{\log r_{j}}{2\log|t|}-b_{j}\right)^{2}-1,$$

on $U\backslash\bigcup_{i\in I}E_{i}$. Choosing the constants $b_{j}$ in the strict interior of $\Delta_{J}$ we can ensure the minimum of $\hat{\rho}$ on $U\cap X_{t}$ equals $-1$, and choosing $A$ suitably large independent of $t$, we can ensure $\{\hat{\rho}\leqslant 0\}$ is compactly contained in $U$. Now we define

$$\rho_{t}=\begin{cases}\min(\hat{\rho},0)|_{X_{t}}\quad&\text{ on }U\cap X_{t}\\ 0\quad&\text{ on }X_{t}\backslash U\end{cases},$$

which satisfies the requirements in (a). We let $B_{t}=\{\rho_{t}<0\}\subset X_{t}$. Lastly, (3.3) follows immediately from part (a) and (3.2). ∎

Thanks to Proposition 3.1, on $B_{t}\subset X_{t}$ we have

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

for some constant $C$ independent of $t$. We then use this together with the elementary inequality $|d\rho_{t}|_{\omega_{t}}^{2}\leqslant|d\rho_{t}|_{\omega_{t}^{\prime}}^{2}\Tr_{\omega_{t}}\omega_{t}^{\prime}$ to get

$$\left(\int_{X_{t}}|d\rho_{t}|_{\omega_{t}}d\mu_{t}\right)^{2}=\left(\int_{B_{t}}|d\rho_{t}|_{\omega_{t}}d\mu_{t}\right)^{2}\leqslant\int_{B_{t}}|d\rho_{t}|_{\omega_{t}}^{2}d\mu_{t}\leqslant C\int_{X_{t}}\Tr_{\omega_{t}}\omega_{t}^{\prime}d\mu_{t},$$

while from (2.1) we get

$$\int_{X_{t}}\Tr_{\omega_{t}}\omega_{t}^{\prime}d\mu_{t}=\frac{n\int_{X_{t}}\omega^{\prime}_{t}\wedge\omega_{t}^{n-1}}{\int_{X_{t}}\omega_{t}^{n}}=\frac{n\int_{X_{t}}c_{1}(\mathfrak{L})\cdot c_{1}(L)^{n-1}}{\int_{X_{t}}c_{1}(L)^{n}}\leqslant C,$$

and so

(3.4)

$$\int_{X_{t}}|d\rho_{t}|_{\omega_{t}}d\mu_{t}\leqslant C.$$

Define two subsets of $X_{t}$ by $A_{1}=\{\rho_{t}<-\frac{2}{3}\}$ and $A_{2}=\{-\frac{1}{3}\leqslant\rho_{t}\leqslant 0\}$. Given two points $x\in A_{1},y\in A_{2}$ which are connected by a unique minimal geodesic $\gamma_{x,y}$ (w.r.t. $\omega_{t}$), we can bound

(3.5)

$$\rho_{t}(y)-\rho_{t}(x)\leqslant\int_{\gamma_{x,y}}|d\rho_{t}|_{\omega_{t}}ds,$$

where $\gamma_{x,y}$ is parametrized with respect to $\omega_{t}$-arclength.

Combining (3.5) with Cheeger-Colding’s segment inequality [4, Theorem 2.11] applied to the function $|d\rho_{t}|_{\omega_{t}}$ we obtain

(3.6)

$$\begin{split}D_{t}(\mu_{t}(A_{1})+\mu_{t}(A_{2}))\int_{X_{t}}|d\rho_{t}|_{\omega_{t}}d\mu_{t}&\geqslant C^{-1}\int_{A_{1}\times A_{2}}\left(\int_{\gamma_{x,y}}|d\rho_{t}|_{\omega_{t}}ds\right)d\mu_{x}d\mu_{y}\\ &\geqslant C^{-1}\int_{A_{1}\times A_{2}}(\rho_{t}(y)-\rho_{t}(x))d\mu_{x}d\mu_{y}\\ &\geqslant\frac{C^{-1}}{3}\mu_{t}(A_{1})\mu_{t}(A_{2}),\end{split}$$

where $D_{t}=\mathrm{diam}(X_{t},\omega_{t})$, and in the $\int_{A_{1}\times A_{2}}$ we are actually only integrating over the subset of pairs $(x,y)$ which are joined by a unique $\omega_{t}$-minimal geodesic, which has full measure (cf. [4]).

Combining (3.4) and (3.6) gives

$$\mu_{t}(A_{1})\mu_{t}(A_{2})\leqslant CD_{t}(\mu_{t}(A_{1})+\mu_{t}(A_{2}))\leqslant CD_{t}.$$

Lastly, from the definition of $\rho_{t}$ and from (3.1), a direct computation in polar coordinates (analogous to the one in [3]) gives

$$\mu_{t}(A_{1})\geqslant C^{-1},\quad\mu_{t}(A_{2})\geqslant C^{-1},$$

for a fixed constant $C$, and so $D_{t}\geqslant C^{-1},$ as desired.

∎