The argument in the previous subsection implies that $T$ maps the smooth set in $X_{\infty}$ to the regular set in $W$. So we need to show that if $T(p)$ is a smooth point of $W$, then the limit metric on $X_{\infty}$ is also smooth at $p$. Denote by $\omega_{i}$ the Kähler-Einstein metric on $X_{i}$, and $\omega_{i}^{\prime}$ the induced Fubini-Study metric. Then we have $\omega_{i}^{\prime}=\omega_{i}+\sqrt{-1}\partial\bar{\partial}\phi_{i}$ with $\phi_{i}=k^{-1}\log\rho_{k}(\omega_{i})$. By our main Theorem 1.1 and Proposition 2.1 there is a constant $C_{1}>0$ such that $|\phi_{i}|_{L^{\infty}}\leq C_{1}$ for all $i$. Also by arguments similar to the proof of Lemma 4.3 we see that there is a constant $C_{2}>0$ such that for all $i$ we have $|\nabla_{\omega_{i}}\phi_{i}|_{L^{\infty}}\leq C_{2}$, and $\omega_{i}^{\prime}\leq C_{2}\omega_{i}$. Now write $Ric(\omega_{i}^{\prime})=\lambda\omega_{i}^{\prime}+\sqrt{-1}\partial\bar{\partial}h_{i}$, where $\lambda$ is $1$, $-\frac{1}{2}$ or $0$. So with suitable normalization of $h_{i}$ we have the equation

(4.1)

$$\omega_{i}^{n}=e^{h_{i}+\lambda\phi_{i}}\omega_{i}^{\prime n}.$$

Then it is not hard to see that $\int_{X_{i}}h_{i}^{2}\omega_{i}^{\prime n}\leq C_{3}$ for some constant $C_{3}>0$. Now for any $p$ in $W^{reg}$, we choose a small neighborhood $B(p,\delta)\subset W^{reg}$. Then there are corresponding points $p_{i}\in X_{i}$, such that $B(p_{i},\delta)$ converges smoothly to $B(p,\delta)$ in $\mathbb{C}\mathbb{P}^{N_{k}}$. By standard elliptic estimate we see that $|h_{i}|_{C^{1}(B(p_{i},\delta/2),\omega_{i}^{\prime})}$ is uniformly bounded. Then by (4.1) there is a $C_{4}>0$ such that $C_{4}^{-1}\omega_{i}\leq\omega_{i}^{\prime}\leq C_{4}\omega_{i}$ in $B(p_{i},\delta/2)$. Thus $|\nabla_{\omega_{i}^{\prime}}\phi_{i}|_{L^{\infty}(B(p_{i},\delta/2))}\leq C_{4}|\nabla_{\omega_{i}}\phi_{i}|_{L^{\infty}(B(p_{i},\delta/2))}\leq C_{4}C_{2}$. Then in $B(p_{i},\delta/2)$ with respect to the metric $\omega_{i}^{\prime}$, the right hand side of (4.1) has a uniform $C^{1}$ bound. Therefore we can apply the Evans-Krylov theory(see for example [3]) to conclude that $|\phi_{i}|$ has a uniform $C^{2,\alpha}$ bound in $B(p_{i},\delta/4)$. Then standard arguments show that all covariant derivatives of $\phi_{i}$(with respect to $\omega_{i}^{\prime}$) are uniformly bounded, so the Kähler-Einstein metrics $\omega_{i}$ converge smoothly in a neighborhood of $p$.

∎

By general theory, what the statement really means is that for any singular point $x$ in $W$, there is a neighborhood $U$, and a nowhere zero holomorphic $n$ form $\Theta$ on $W^{reg}\cap U$ with $\int_{W^{reg}\cap U}\Theta\wedge\overline{\Theta}<\infty$. We first consider the cases $L=K_{X}^{2}$ and $K_{X}^{-1}$. Previous discussion has shown that for any $x$, there is a neighborhood $U$ of $x$, an integer $k>0$, a constant $C>0$, and a section $s$ of $L^{k}$ over $X_{\infty}\setminus\Sigma=W^{reg}$ with $C^{-1}\leq\|s(x)\|^{2}\leq C$ for $x\in W^{reg}\cap U$. Here the norm is taken with respect to the Kähler-Einstein metric. When $L=K_{X}^{2}$, we define $\Theta=(s\otimes\overline{s})^{\frac{1}{2k}}$, then

$$\int_{W^{reg}\cap U}\Theta\wedge\overline{\Theta}=\int_{W^{reg}\cap U}\|s\|^{\frac{1}{k}}dvol\leq C^{\frac{1}{2k}}Vol(W).$$

When $L=-K_{X}$, we define $\Theta=(s^{*}\otimes\overline{s^{*}})^{\frac{1}{k}}$, where $s^{*}$ is the dual section of $s$. So $\|s^{*}\|=\|s\|^{-1}$. Then

$$\int_{W^{reg}\cap U}\Theta\wedge\overline{\Theta}=\int_{W^{reg}\cap U}\|s^{*}\|^{\frac{2}{k}}dvol\leq C^{-\frac{1}{k}}Vol(W).$$

In the Calabi-Yau case, since $\Theta_{i}$ has norm one, we easily see that there is a limit holomorphic volume form $\Theta$ on $X_{\infty}\setminus\Sigma=W^{reg}$ with norm one. Then $\int_{W^{reg}}\Theta\wedge\overline{\Theta}=Vol(W).$ ∎