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

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 $p$.

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