We plan to exhibit a $T^{2}$-bundle preserving diffeomorphism $\Psi_{1}$ between the Taub-NUT type $\mathbb{C}^{3}$ and the positive vertex space $M^{+}_{\nu}$ over the common base $\{0<|\vec{\mu}|_{a}\leq\frac{1}{3}A^{1/2}\}$, with good estimates on the deviations between both Kähler structures. Since $\frac{1}{3}A^{1/2}<\frac{1}{2}A^{1/2}$, the $\eta$-periodic copies of such punctured discs do not overlap. The topology of the $T^{2}$-bundle structures on both spaces agree by construction. The remaining degrees of freedom in defining $\Psi_{1}$ amounts to a gauge choice, which is the same as a prescription of $\vartheta_{i}^{\mathbb{C}^{3}}-\Psi_{1}^{*}\vartheta_{i}$.

As a general guideline, the corresponding quantities on $\mathbb{C}^{3}$ and $M^{+}_{\nu}$ have the same singularity, so their difference are smooth quantities. We use superscripts for quantities on $\mathbb{C}^{3}$ to disambiguate from quantities on $M^{+}_{\nu}$.

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 $\tilde{Z}_{i}$.

We make a second modification from $(\tilde{g}^{(3)},\tilde{\omega}^{(3)},\Omega)$ to another new Kähler ansatz $(\tilde{g}^{(4)},\tilde{\omega}^{(4)},\Omega)$ designed to match up with the Taub-NUT type metric $(g_{\mathbb{C}^{3}},\omega_{\mathbb{C}^{3}},\Omega_{\mathbb{C}^{3}})$ in Chapter 2.

Recall from Theorem 2.26 that there is a Kähler potential $\phi^{\mathbb{C}^{3}}$ such that

$$\omega_{\mathbb{C}^{3}}=\omega^{(2)}+\sqrt{-1}\partial\bar{\partial}\phi^{\mathbb{C}^{3}}=\omega^{(1)}+\sqrt{-1}\partial\bar{\partial}\phi^{\mathbb{C}^{3}},\quad\text{for }|\vec{\mu}|_{a}\gtrsim A^{-1/4},$$

with bound $\left\lVert d\phi^{\mathbb{C}^{3}}\right\rVert_{C^{k+1,\alpha}_{-\epsilon,-1+\epsilon}(\mathbb{C}^{3})}\leq CA^{-1/4}.$ We can impose a normalisation such that $|\phi^{\mathbb{C}^{3}}|\leq CA^{-1/2+\frac{3}{4}\epsilon}$ for $\frac{1}{100}A^{1/2}\lesssim|\vec{\mu}|_{a}\leq\frac{1}{3}A^{1/2}.$

We then define a modified Kähler metric ansatz $\tilde{\omega}^{(4)}$ on $M^{+}_{\nu}$. Take a standard cutoff function

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

and define

$$\tilde{\omega}^{(4)}=\tilde{\omega}^{(3)}+\sqrt{-1}\partial\bar{\partial}\phi_{4},\quad\phi_{4}=\chi(\frac{|\vec{\mu}|_{a}}{\frac{1}{12}A^{1/2}})\phi^{\mathbb{C}^{3}}-2\varphi_{3}.$$

In particular

$$\begin{cases}\tilde{\omega}^{(4)}=\omega_{\mathbb{C}^{3}}-2\sqrt{-1}\partial\bar{\partial}\varphi_{3},\quad&|\vec{\mu}|_{a}\leq\frac{1}{12}A^{1/2},\\ \tilde{\omega}^{(4)}=\tilde{\omega}^{(3)},\quad&|\vec{\mu}|_{a}\geq\frac{1}{3}A^{1/2}.\end{cases}$$

The positive definiteness of $\tilde{\omega}^{(4)}$ follows from the metric deviation estimate:

$$\begin{cases}\left\lVert\partial\bar{\partial}\varphi_{3}\right\rVert_{C^{k,\alpha}_{0,0}(\mathbb{C}^{3}\cap\{|\vec{\mu}|_{a}\leq\frac{1}{3}A^{1/2}\})}\leq C\nu A^{-3/4},\\ \left\lVert\partial\bar{\partial}\{\chi(\frac{|\vec{\mu}|_{a}}{\frac{1}{12}A^{1/2}})\phi^{\mathbb{C}^{3}}\}\right\rVert_{C^{k,\alpha}_{-1-\epsilon,0}(|\vec{\mu}|_{a}\sim A^{1/2})}\leq CA^{3/4(-1+\epsilon)}.\end{cases}$$

Here $\varphi_{3}$ is inserted to approximately cancel the cutoff error $\Delta_{a}\varphi_{3}$ in the volume form error $\tilde{E}^{(3)}$ (cf. Lemma 3.28).

Now we introduce the global weighted Hölder norms $\left\lVert\cdot\right\rVert_{C^{k,\alpha}_{\delta}}$ on $M^{+}_{\nu}$ by demanding that up to uniform equivalence the norm is

• •

  $\left\lVert\cdot\right\rVert_{C^{k,\alpha}_{\delta}}$ on $M^{+}\cap\{|\vec{\mu}|_{a}\gtrsim A^{-1/4}\}$, as defined in Section 3.4.

• •

  $\left\lVert\cdot\right\rVert_{C^{k,\alpha}_{\delta,0}(\mathbb{C}^{3}\cap\{|\vec{\mu}|_{a}\leq\frac{1}{3}A^{1/2}\})}$ on $\mathbb{C}^{3}$ for $|\vec{\mu}|_{a}\leq\frac{1}{3}A^{1/2}$.

On overlapping regions the definitions are equivalent.