We now modify $(\tilde{g}^{(2)},\tilde{\omega}^{(2)},\tilde{\Omega}^{(2)})$ to an intermediate Kähler ansatz $(\tilde{g}^{(3)},\tilde{\omega}^{(3)},\Omega)$ designed to match up exactly with $(g^{(1)},\omega^{(1)},\Omega_{\mathbb{C}^{3}})$ over $\{|\vec{\mu}|_{a}\leq\frac{1}{6}A^{1/2}\}$. This will be constructed using the generalised Gibbons-Hawking ansatz.

Take a standard cutoff function $\chi$ on $\mathbb{R}$ with

$$\chi(s)=\begin{cases}1\quad s\leq 1,\\ 0\quad s\geq 2,\end{cases}$$

and let $\varphi_{3}=\chi(\frac{|\vec{\mu}|_{a}}{\frac{1}{6}A^{1/2}})\varphi_{2}$ with $\varphi_{2}$ from Corollary 3.27,

$$\tilde{V}_{(3)}^{ij}=\tilde{V}_{(2)}^{ij}+\frac{\partial^{2}\varphi_{3}}{\partial\mu_{i}\partial\mu_{j}},\quad\tilde{W}_{(3)}^{ij}=\tilde{W}_{(2)}^{ij}-4\frac{\partial^{2}\varphi_{3}}{\partial\eta\partial\bar{\eta}}.$$

The perturbations are sufficiently small so that positive definiteness is not affected. The generalised Gibbons-Hawking construction produces the intermediate Kähler ansatz $(\tilde{g}^{(3)},\tilde{\omega}^{(3)},\Omega)$. We identify $M^{+}_{\nu}$ with the underlying space of $(\tilde{g}^{(3)},\tilde{\omega}^{(3)},\Omega)$. The $T^{2}$-connection for $(\tilde{g}^{(3)},\tilde{\omega}^{(3)},\Omega)$ is identified as (cf. (1.13))

$$\vartheta_{i}^{(3)}=\vartheta_{i}^{(2)}+\sqrt{-1}\frac{\partial^{2}\varphi_{3}}{\partial\eta\partial\mu_{i}}d\eta-\sqrt{-1}\frac{\partial^{2}\varphi_{3}}{\partial\bar{\eta}\partial\mu_{i}}d\bar{\eta}.$$

This amounts to making a gauge choice.

By construction $(\tilde{g}^{(3)},\tilde{\omega}^{(3)},\Omega)$ agrees identically with $(\tilde{g}^{(2)},\tilde{\omega}^{(2)},\tilde{\Omega}^{(2)})$ over $\{|\vec{\mu}|_{a}\geq\frac{1}{3}A^{1/2}\}$, and modulo diffeomorphism agrees identically with $({g}^{(1)},{\omega}^{(1)},\Omega_{\mathbb{C}^{3}})$ over $\{|\vec{\mu}|_{a}\leq\frac{1}{6}A^{1/2}\}$. By Corollary 3.27,

Henceforth the complex structure will be fixed, and can be identified as follows. The new holomorphic differentials are

(3.10)

$$\begin{cases}d\log\tilde{Z}_{i}=d\log Z_{i}+d(\frac{\partial(\varphi_{3}+\varphi_{1})}{\partial\mu_{i}}),\quad i=1,2,\\ d\log\tilde{Z}_{0}=d\log Z_{0}-d(\frac{\partial(\varphi_{3}+\varphi_{1})}{\partial\mu_{1}}+\frac{\partial(\varphi_{3}+\varphi_{1})}{\partial\mu_{2}}).\end{cases}$$

These have the same $T^{3}$-periods as $d\log Z_{i}$, which lie inside $2\pi\sqrt{-1}\mathbb{Z}$, so the new holomorphic functions $\tilde{Z}_{0},\tilde{Z}_{1},\tilde{Z}_{2}$ are defined without multivalue issues. The functional equation

$$\tilde{Z}_{0}\tilde{Z}_{1}\tilde{Z}_{2}=1-e^{2\pi\sqrt{-1}\eta}=1-Z_{3}$$

persists from Lemma 3.10. The results in Proposition 3.11 hold verbatim:

Over $\{|\vec{\mu}|_{a}\leq\frac{1}{6}A^{1/2}\}$ the ansatz $(\tilde{g}^{(3)},\tilde{\omega}^{(3)},\Omega)$ is identified with $({g}^{(1)},\omega^{(1)},\Omega_{\mathbb{C}^{3}})$ after suitable diffeomorphism. An identification of complex coordinates compatible with the holomorphic differential formula (3.10) is

$$\begin{cases}\tilde{Z}_{1}=z_{1}\exp\{(\pi\sqrt{-1}\beta_{1}(0,0,1)-\sqrt{-1}\theta_{1}^{\infty})\eta\},\\ \tilde{Z}_{2}=z_{2}\exp\{(\pi\sqrt{-1}\beta_{2}(0,0,1)-\sqrt{-1}\theta_{2}^{\infty})\eta\},\\ \tilde{Z}_{0}=\frac{-2\sqrt{-1}\sin(\pi\eta)}{\eta}z_{0}\exp\{(\pi\sqrt{-1}\beta_{0}(0,0,1)+\sqrt{-1}\theta_{1}^{\infty}+\sqrt{-1}\theta_{2}^{\infty})\eta\}.\end{cases}$$

This fixes the normalisation for the multiplicative constants of $\tilde{Z}_{i}$.