We choose a finite cover $D=\bigcup_{k=1}^{N_{0}}O_{k}$ such that for each $k$ there exists a local holomorphic coordinate system $(z,w)$ on some domain $U_{k}\subset M$ such that $U_{k}\cap D=O_{k}=\{w=0\}$. We will show that $\bar{\partial}\beta_{0}=0$ in every $O_{k}\subset D$ in the distributional sense. Let $\psi$ be a smooth section of $\Lambda^{0,1}_{D}\otimes(K_{M}^{-1}|_{D})$ with compact support in $O_{k}$. It suffices to show that $\langle\beta_{0},\bar{\partial}^{*}\psi\rangle_{O_{k}}=0$. To this end, write $\psi(z)=\sigma(z)d\overline{z}\otimes(dz\wedge dw)^{-1}$ for some smooth function $\sigma\in C^{\infty}_{0}(O_{k},\mathbb{C})$ and use this to define the trivial extension $\hat{\psi}(z,w)=\sigma(z)d\overline{z}\otimes(dz\wedge dw)^{-1}$ for all $(z,w)\in U_{k}$. Denote by $O_{k}(\tau)$ the slice $\{w=\tau\}$ in $U_{k}$, which is a complex submanifold of $M$, and equip $O_{k}(\tau)$ with the restriction of the Kähler metric $\omega_{h}$ from $M$. Notice that $\hat{\psi}$ restricts to a smooth section of $\Lambda_{O_{k}(\tau)}^{0,1}\otimes(K_{M}^{-1}|_{O_{k}(\tau)})$ with compact support in $O_{k}(\tau)$. Since $\bar{\partial}\beta=\alpha$ and $\beta\in W^{1,p}\cap C^{\alpha}$ for any $p\geq 1$, it follows that

(5.10)

$\displaystyle\langle\beta_{0},\bar{\partial}^{*}{\psi}\rangle_{O_{k}}=\lim\limits_{\tau\to 0}\langle\beta,\bar{\partial}^{*}\hat{\psi}\rangle_{O_{k}(\tau)}=\lim\limits_{\tau\to 0}\langle\bar{\partial}\beta,\hat{\psi}\rangle_{O_{k}(\tau)}=\lim\limits_{\tau\to 0}\langle\alpha,\hat{\psi}\rangle_{O_{k}(\tau)}.$

Notice that

(5.11)

$$|\gamma(\partial_{\bar{z}})|\leq|\gamma|_{\omega_{TY}}|\partial_{\bar{z}}|_{\omega_{TY}}\leq|\gamma|_{\omega_{TY}}(-\log|S|^{2}_{h})^{\frac{1}{4}}=O(|S|_{h}^{-\epsilon}).$$

Since $\alpha=\gamma\otimes S$, it then follows that $|\alpha(\partial_{\bar{z}})|=O(|S|_{h}^{1-\epsilon})\rightarrow 0$ uniformly as $w\rightarrow 0$. Using (5.10), it follows that

(5.12)

$$\langle\beta_{0},\bar{\partial}^{*}{\psi}\rangle_{O_{k}}=0,$$

as desired. By standard elliptic regularity, $\beta_{0}$ is a holomorphic section. ∎