In Section 4.2 we have identified the underlying complex manifold of our family of $C^{2,\alpha}$ Kähler metrics $(\mathcal{M}_{T},\omega_{T})$. In our gluing argument in Section 7.3 we shall need to perturb the complex structure. This section is devoted to the estimate of error caused by such a perturbation.

Under the holomorphic embedding of $\mathcal{M}_{T}$ into $\mathcal{N}^{0}$ defined in Section 4.2, $\Omega_{T}$ is identified with the standard holomorphic volume form $\Omega_{\mathcal{N}^{0}}$.

Fix $C>0$, and let $\mathcal{V}$ be the open neighborhood of $\mathcal{P}$ in $\mathcal{N}^{0}$ defined by $\{r_{+}<C,r_{-}<C\}$. Fix a smooth Kähler metric $\omega_{\mathcal{N}^{0}}$ on $\mathcal{N}^{0}$. Suppose now that we have a family of complex structures $J_{T}^{\prime}$ on $\mathcal{V}$ with holomorphic volume forms $\Omega_{T}^{\prime}$ satisfying for all $k\geq 0$,

(4.320)

$$\sup_{\bm{x}\in\mathcal{V}}|\nabla^{k}_{\omega_{\mathcal{N}^{0}}}(\Omega_{T}^{\prime}-\Omega_{\mathcal{N}^{0}})(\bm{x})|_{\omega_{\mathcal{N}^{0}}}\leq\underline{\epsilon}_{T^{2}}.$$

We also assume there is a deformation of the form $\pi^{*}\omega_{D}$ over $\mathcal{V}$ to $\omega_{D,T}$, which is a closed $(1,1)$ form with respect $J_{T}^{\prime}$, and satisfies that for all $k\geq 0$

(4.321)

$$\sup_{\bm{x}\in\mathcal{V}}|\nabla^{k}_{\omega_{\mathcal{N}^{0}}}(\omega_{D,T}-\pi^{*}\omega_{D})(\bm{x})|_{\omega_{\mathcal{N}^{0}}}\leq\underline{\epsilon}_{T^{2}}.$$

Let $\phi$ be the Kähler potential defined in (4.149). Then we define the new family of closed forms on $\mathcal{V}$

(4.322)

$$T^{\frac{n-2}{n}}\omega_{T}^{\prime}\equiv T\omega_{D,T}+dJ_{T}^{\prime}d\phi.$$

We first reduce the estimate to a local form. Choose finitely many holomorphic charts $(U_{\beta},w_{1},\cdots,w_{n-1})$ in $D$ of the form $\{|w_{1}|<1,\cdots,|w_{n-1}|<1\}$, such that the smaller charts given by $V_{\beta}=\{|w_{1}|<1/2,\cdots,|w_{n-1}|<1/2\}$ also cover $D$. We may also assume if a $U_{\beta}$ intersects $H$, then it is centered at some $p\in H$, i.e. $w_{i}(p)=0$ for all $i$, and also $H$ is defined by $w_{1}=0$ in this chart. We may further assume in each $U_{\beta}$ the line bundle $L$ has a holomorphic trivialization $\sigma_{\beta}$, under which we may view $\zeta_{\pm}$ as local holomorphic functions on $\mathcal{N}^{0}$, and $\mathcal{V}\cap\pi^{-1}(U_{\beta})$ is locally defined by $|\zeta_{\pm}|<C|\sigma_{L}|^{-|k_{\pm}|}$. These then give an open cover of $\mathcal{V}$ by $\mathcal{V}\cap\pi^{-1}(V_{\beta})$, and it suffices to prove the estimates in each such open set.

We shall work with one $\beta$ such that $U_{\beta}\cap H\neq\emptyset$. The other case can be proved similarly. For such $\beta$ in $\pi^{-1}(U_{\beta})$, by definition of $\mathcal{N}^{0}$, we get the equation

(4.325)

$$\zeta_{+}\cdot\zeta_{-}=w_{1}F(w_{1},\cdots,w_{n-1})$$

for a non-zero holomorphic function $F$. So without loss of generality we may assume $\zeta_{+},\zeta_{-},w_{2},\cdots,w_{n-1}$ are holomorphic coordinates on $\pi^{-1}(U_{\beta})$.

We first prove (4.323). The hypothesis implies that

(4.326)

$$\Omega_{T}^{\prime}-\Omega_{T}=G_{i_{1}\ldots i_{n}}e_{i_{1}}\wedge\ldots\wedge e_{i_{n}},$$

where each $e_{j}$ is one of $d\zeta_{\pm},d\bar{\zeta}_{\pm},dw_{j},d\bar{w}_{j}(j\geq 2)$, and $G_{i_{1}\ldots i_{n}}$ is a smooth function in $\zeta_{\pm},\bar{\zeta}_{\pm},w_{j},\bar{w}_{j}(j\geq 2)$ and its $k$-th derivative over $\mathcal{V}$ with respect to the fixed metric $\omega_{\mathcal{N}^{0}}$ is bounded by $\underline{\epsilon}_{T^{2}}$ for all $k$. Since a holomorphic function is automatically harmonic with respect to any Kähler metric, we have

(4.327)

$$\Delta_{\omega_{T}}\zeta_{\pm}=\Delta_{\omega_{T}}w_{j}=0.$$

By Corollary 4.11.1, Item (1), we know $\mathcal{V}$ is contained in the region $|z|\leq 1$. Also notice by the discussion in Section 4.3 there is a constant $C>1$ such that for each $\bm{x}\in\mathcal{V}\cap\pi^{-1}(V_{\beta})$, the ball $B_{C^{-1}\mathfrak{s}(\bm{x})}(\bm{x})$ is contained in $\pi^{-1}(U_{\beta})\cap\{|z|\leq 2\}$. Again by Corollary 4.11.1, Item (3) on $\pi^{-1}(U_{\beta})\cap\{|z|\leq 2\}$, we have

(4.328)

$$|\zeta_{\pm}|\leq Cr_{\pm}\leq Ce^{3T}.$$

Now applying Proposition 4.22 to every $\bm{x}\in\mathcal{V}\cap\pi^{-1}(V_{\beta})$ we obtain

(4.329)

$$|\zeta_{\pm}|_{C^{3,\alpha}_{\delta,\mu,\nu}(\mathcal{V})}\leq C|\zeta_{\pm}|_{C^{0}_{\delta,\mu,\nu}(\pi^{-1}(U_{\beta})\cap\{|z|\leq 1\})}\leq Ce^{3T}.$$

Similarly, since on $U_{\beta}$ we have $|w_{j}|<1$, we get for $j=2,\cdots,n-1$,

(4.330)

$$|w_{j}|_{C^{3,\alpha}_{\delta,\mu,\nu}(\mathcal{V})}\leq|w_{j}|_{C^{0}_{\delta,\mu,\nu}(\pi^{-1}(U_{\beta})\cap\{|z|\leq 1\})}\leq C.$$

Then using the chain rule and induction we get that

(4.331)

$$|G_{i_{1}\cdots i_{n}}|_{C^{3,\alpha}_{\delta,\nu,\mu}(\mathcal{V})}=\underline{\epsilon}_{T^{2}}\cdot Ce^{3T}=\underline{\epsilon}_{T^{2}}.$$

So

(4.332)

$$|\Omega_{T}^{\prime}-\Omega_{T}|_{C^{2,\alpha}_{\delta,\mu,\nu}(\mathcal{V})}=\underline{\epsilon}_{T^{2}}.$$

Notice the complex structure $J_{T}^{\prime}$ is pointwise determined by the holomorphic $n$ form $\Omega_{T}^{\prime}$ algebraically, we get

(4.333)

$$|J_{T}^{\prime}-J_{T}|_{C^{2,\alpha}_{\delta,\nu,\mu}(\mathcal{V})}=\underline{\epsilon}_{T^{2}}.$$

Now to prove (4.324), we write

(4.334)

$$\omega_{T}^{\prime}=\omega_{T}+T(\omega_{D,T}-\pi^{*}\omega_{D})+d((J_{T}^{\prime}-J_{T})d\phi).$$

By assumption, and the above discussion, using (4.329) we get

(4.335)

$$|\omega_{D,T}-\pi^{*}\omega|_{C^{2,\alpha}_{\delta,\nu,\mu}(\pi^{-1}(U_{\beta})\cap\{|z|\leq 1\})}=\underline{\epsilon}_{T^{2}}$$

It is also easy to see

(4.336)

$$|V_{1}\cdot V_{2}|_{C^{2,\alpha}_{\delta,\nu,\mu}(\mathcal{V})}\leq T^{m}|V_{1}|_{C^{2,\alpha}_{\delta,\nu,\mu}(\mathcal{V})}|V_{2}|_{C^{2,\alpha}_{\delta,\nu,\mu}(\mathcal{V})},$$

for some $m>0$ independent of $V_{1}$ and $V_{2}$. So (4.324) is a consequence of the following