Let $\mu_{1},\mu_{2}$ be fixed. The essential task is to understand the asymptotic behaviour of $\beta_{i}(\mu_{1},\mu_{2},\eta+n)$ as $|n|$ becomes large. We focus on $\beta_{1}$.

Using the homogeneity property of $\alpha_{1},\alpha_{2},\alpha_{3}$ in the $\mu_{1},\mu_{2}$ and $\eta$ variables, it is easy to see from the integral definition of $\beta_{1}$ that

$$\beta_{1}(0,0,\eta)=\frac{1}{\eta}\beta_{1}(0,0,1),\quad|\beta_{1}(0,0,1)|\leq C.$$

By elementary properties of arctan

$$\alpha_{1}(\mu_{1},\mu_{2},\eta)=\frac{1}{4\sqrt{\mu_{1}^{2}+a_{22}|\eta|^{2}}}+O(\frac{|\mu_{1}|+|\mu_{2}|}{\mu_{1}^{2}+a_{22}|\eta|^{2}}),$$

$$\frac{\partial\alpha_{1}}{\partial\eta}=\frac{-a_{22}\bar{\eta}}{8(\mu_{1}^{2}+a_{22}|\eta|^{2})^{3/2}}\left(1+O(\frac{|\mu_{1}|+|\mu_{2}|}{(\mu_{1}^{2}+a_{22}|\eta|^{2})^{1/2}})\right),$$

and similarly

$$\frac{\partial\alpha_{3}}{\partial\eta}=\frac{-(a_{11}+2a_{22}+a_{22})\bar{\eta}}{8((\mu_{1}-\mu_{2})^{2}+(a_{11}+2a_{22}+a_{22})|\eta|^{2})^{3/2}}\left(1+O(\frac{|\mu_{1}|+|\mu_{2}|}{((\mu_{1}-\mu_{2})^{2}+A^{1/2}|\eta|^{2})^{1/2}})\right).$$

After integration

$$\begin{split}|\beta_{1}(\mu_{1},\mu_{2},\eta)-\beta_{1}(0,0,\eta)|\leq\frac{C(|\mu_{1}|+|\mu_{2}|)}{|\eta|}(\frac{1}{\sqrt{\mu_{1}^{2}+a_{22}|\eta|^{2}}}+\frac{1}{\sqrt{(\mu_{1}-\mu_{2})^{2}+a_{22}|\eta|^{2}}}).\end{split}$$

This shows the series

$$\sum_{n\in\mathbb{Z}}\beta_{1}(\mu_{1},\mu_{2},\eta+n)-\beta_{1}(0,0,\eta+n)$$

is absolutely convergent if $\eta\notin\mathbb{Z}$, and if morever $|x|\leq\frac{1}{2}$ then we have the bound

$$\sum_{n\in\mathbb{Z}\setminus\{0\}}|\beta_{1}(\mu_{1},\mu_{2},\eta+n)-\beta_{1}(0,0,\eta+n)|\leq\frac{C(|\mu_{1}|+|\mu_{2}|)}{A^{1/4}}.$$

Thus the convergence of the series $\tilde{\beta}_{1}$ is equivalent to the convergence of

$$\lim_{N\to\infty}\sum_{n=-N}^{N}\beta_{1}(0,0,\eta+n)=\beta_{1}(0,0,1)\lim_{N\to\infty}\sum_{n=-N}^{N}\frac{1}{\eta+n}=\beta_{1}(0,0,1)\pi\cot(\pi\eta),$$

and similarly for $\tilde{\beta}_{2}$ and $\tilde{\beta}_{0}$. The periodicity claim follows from standard rearranging theorems for series. The estimate on $\tilde{\beta}_{i}-\beta_{i}$ follows by combining the above discussions. ∎

This is the periodic version of Lemma 2.7. The terms $\beta_{i}(0,0,1)\pi\sqrt{-1}$ and $\theta_{i}^{\infty}$ are added for later convenience. ∎

This is the periodic version of Lemma 2.8, using Euler’s series identity for $\cot(\pi\eta)$:

$$\lim_{N\to\infty}\sum_{n=-N}^{N}\frac{1}{\eta+n}=\pi\cot(\pi\eta),$$

and $\sum_{i}\beta_{i}(0,0,1)=1$ from Lemma 2.8. ∎

The periods along the generating cycles in the $T^{2}$-fibres are straightforward:

$$\int_{S^{1}}\tilde{\zeta}_{i}^{\prime}=\int_{S^{1}}\sqrt{-1}\vartheta_{i}=2\pi\sqrt{-1},\quad i=1,2,$$

and $\int_{S^{1}}\tilde{\zeta}_{0}^{\prime}=-\sqrt{-1}\int_{S^{1}}\vartheta_{1}+\vartheta_{2}=-4\pi\sqrt{-1}$.

Computing the period along the other $S^{1}$ requires a special trick. As a preparatory subtle remark, the Kähler metric is not globally defined over the base $\mathbb{R}^{2}_{\mu_{1},\mu_{2}}\times(S^{1}\times\mathbb{R})_{\eta}$ due to incompleteness issues, but the quantities $\tilde{v}^{ij},\tilde{w},\vartheta_{i}$ make sense globally. Consider the $S^{1}$ on the base defined by $\{\mu_{1}=\mu_{2}=0,y=\text{const}\}$. If we attempt to lift this $S^{1}$ by parallel transport, in general we cannot get a closed loop, and this failure is measured by the holonomy of the $T^{2}$-connection $\vartheta=(\vartheta_{1},\vartheta_{2})$ along the $S^{1}$. When $y\to+\infty$, due to the exponential decay of the $x$-dependent part of $\tilde{v}^{ij},\tilde{w},\vartheta_{i}$, this holonomy converges to two real numbers $(\theta_{1}^{\infty},\theta_{2}^{\infty})$ modulo $2\pi\mathbb{Z}$. In particular, if we twist $\vartheta$ by a flat $T^{2}$-connection, then $\theta_{1}^{\infty},\theta_{2}^{\infty}$ receive a corresponding twist so that $\tilde{\zeta}_{i}$ is unaffected. Thus we can assume without loss of generality that $\theta_{i}^{\infty}=0$, namely the asymptotic holonomy of $\vartheta$ is zero, so in the limit the $S^{1}$ cycle lifts to a closed loop, on which we can evaluate the period asymptotically.

By construction $\int_{S^{1}}\vartheta_{i}=0$, and using $\tilde{\beta}_{i}(0,0,\eta)=\beta_{i}(0,0,1)\pi\cot(\pi\eta)$ from the proof of Lemma 3.7, we compute

$$\begin{split}\int_{S^{1}}\tilde{\zeta}_{i}^{\prime}&=\int_{S^{1}}(\tilde{\beta}_{i}+\beta_{i}(0,0,1)\pi\sqrt{-1})d\zeta\\ &=\lim_{y\to\infty}\int_{S^{1}}\beta_{i}(0,0,1)(\pi\cot(\pi\eta)+\pi\sqrt{-1})d\eta\\ &=\lim_{y\to\infty}\beta_{i}(0,0,1)\int_{S^{1}}d\log(1-e^{2\pi\sqrt{-1}\eta})=0.\end{split}$$

From this we see the integrality condition on the periods, so the holomorphic functions $Z_{i}$ are well defined without multivalue issues.

Notice the definition of $Z_{i}$ for $i=0,1,2$ involve three unspecified multiplicative constants; by prescribing their product appropriately, the functional equation follows from Lemma 3.9. The remaining two free multiplicative constants will be fixed in later Sections. ∎

The $T^{2}$-action follows the same argument as Proposition 2.11. The holomorphic volume form is characterised by $\tilde{\Omega}^{(1)}(\frac{\partial}{\partial\theta_{1}},\frac{\partial}{\partial\theta_{2}},\cdot)=d\eta.$ Notice also

$$-dZ_{0}\wedge dZ_{1}\wedge dZ_{2}(\frac{\partial}{\partial\theta_{1}},\frac{\partial}{\partial\theta_{2}},\cdot)=d(Z_{0}Z_{1}Z_{2})=-dZ_{3}=-2\pi\sqrt{-1}Z_{3}d\eta,$$

so $\tilde{\Omega}^{(1)}=-\frac{\sqrt{-1}}{2\pi Z_{3}}dZ_{0}\wedge dZ_{1}\wedge dZ_{2}$. This formula in particular implies the map $M^{+}\to\{Z_{0}Z_{1}Z_{2}=1-Z_{3}\}$ is a local biholomorphism. We finally need to show this map is injective. Since both $M^{+}$ and $\{Z_{0}Z_{1}Z_{2}=1-Z_{3}\}$ fibre over the $\mathbb{C}^{*}$ coordinate $\eta$ in a compatible way, it suffices to compare the fibres, which have compatible $T^{2}$-actions, so boils down to the injectivity of $(\mu_{1},\mu_{2})\mapsto(\log|Z_{0}|,\log|Z_{1}|,\log|Z_{2}|)$ for fixed $\eta$. ∎