As a hyperKähler metric, the Ooguri-Vafa metric admits a 2-sphere of compatible integrable complex structures. There is one distinguished complex structure giving rise to a holomorphic elliptic fibration, which is described in detail in [11]. Here we wish to focus on another distinguished complex structure where $\eta$ is holomorphic and $\mu$ is the symplectic moment map, which is more natural for special Lagrangian fibrations. The author is not aware of explicit references for the content of this Section.

Our main goal is to identify the complex structure on $M$ explicitly, which requires us to construct holomorphic functions on $M$. We start with the type $(1,0)$ form $\zeta=Vd\mu+\sqrt{-1}\vartheta$ and recall formula (1.14). To turn $\zeta$ into a holomorphic differential, we need to subtract a function times $d\eta$, whose differential cancels out $d\zeta$. Inspired by the Taub-NUT example 1.8, and taking care of periodicity requirement, we introduce the functions

$$\begin{cases}\beta_{+}=\frac{\pi\sqrt{-1}}{2}-\sqrt{-1}\theta_{\infty}+\lim_{k\to\infty}\sum_{n=-k}^{n=k}\{\frac{1}{2(\eta+n)}-\frac{\mu}{2(\eta+n)\sqrt{\mu^{2}+|\eta+n|^{2}}}\}\\ \beta_{-}=\frac{\pi\sqrt{-1}}{2}+\sqrt{-1}\theta_{\infty}+\lim_{k\to\infty}\sum_{n=-k}^{n=k}\{\frac{1}{2(\eta+n)}+\frac{\mu}{2(\eta+n)\sqrt{\mu^{2}+|\eta+n|^{2}}}\}\end{cases}$$

These series are convergent and 1-periodic in $\eta$, such that the forms $\zeta^{\prime}=\zeta+\beta_{+}d\eta$, $\zeta^{\prime\prime}=-\zeta+\beta_{-}d\eta$ are closed. The real number $\theta_{\infty}$ is chosen to cancel the asymptotic holonomy of the $S^{1}$-connection $\vartheta$ along the $\text{Re}(\eta)$-circle as $\text{Im}\eta\to\infty$. We have

$$\begin{split}\zeta^{\prime}+\zeta^{\prime\prime}&=\pi\sqrt{-1}d\eta+\lim_{k\to\infty}\sum_{n=-k}^{k}\frac{1}{\eta+n}d\eta\\ &=\pi\sqrt{-1}d\eta+d\log(\eta\prod_{n=1}^{\infty}(1-\frac{\eta^{2}}{n^{2}}))\\ &=\pi\sqrt{-1}d\eta+d\log(\frac{\sin(\pi\eta)}{\pi})=d\log(1-e^{2\pi i\eta}),\end{split}$$

where we made use of Euler’s factorisation identity of $\frac{\sin(\pi\eta)}{\pi\eta}$. The line integrals $\int\zeta^{\prime}$ and $\int\zeta^{\prime\prime}$ are locally holomorphic functions on $M$, defined over the complement of $\{\mu\leq 0,\eta=0\}$ and $\{\mu\geq 0,\eta=0\}$ inside $\mathcal{B}$. Their $T^{2}$-periods lie in $2\pi\sqrt{-1}\mathbb{Z}$: the periods along the $S^{1}$-fibre over $\mathcal{B}$ is $\int_{S^{1}}\sqrt{-1}\vartheta=2\pi\sqrt{-1}$, while the periods along the $S^{1}$-cycle in $M$ lifting $S^{1}\subset\mathbb{R}\times S^{1}\times\mathbb{R}$ can be evaluated by their asymptotic value as $\text{Im}(\eta)\to+\infty$; in particular if we twist the connection $\vartheta$ by a flat connection, then we can always use the choice of $\theta_{\infty}$ to cancel that twist. Thus we can define the holomorphic functions without multivalue issues

$$z_{1}=\exp(\int\zeta^{\prime}),\quad z_{2}=\exp(\int\zeta^{\prime\prime}).$$

We are free to choose the multiplicative constants on $z_{1},z_{2}$ to satisfy the functional equation $z_{1}z_{2}=1-e^{2\pi i\eta}$, from which we see that $z_{1},z_{2}$ extend to global holomorphic functions on $M$, with zero locus $\{\mu\leq 0,\eta=0\}$ and $\{\mu\geq 0,\eta=0\}$ inside $\mathcal{B}$ respectively. Setting $z_{3}=\exp(2\pi\sqrt{-1}\eta)$, we obtain a holomorphic map

$$M\to\{z_{1}z_{2}=1-z_{3}\}\subset\mathbb{C}^{2}_{z_{1},z_{2}}\times\mathbb{C}^{*}_{z_{3}},$$

which is easily seen to be an open embedding.

By looking at the action of the Hamiltonian vector field $\frac{\partial}{\partial\theta}$, we can identify the circle action as

$$e^{i\theta}\cdot(z_{1},z_{2},z_{3})=(e^{i\theta}z_{1},e^{-i\theta}z_{2},z_{3}).$$

By construction the holomorphic volume form $\Omega$ satisfies $\iota_{\frac{\partial}{\partial\theta}}\Omega=d\eta$, which implies $\Omega=-\frac{1}{2\pi}\frac{1}{z_{3}}dz_{1}\wedge dz_{2},$ or equivalently

$$\Omega\wedge d((z_{1}z_{2})^{-1}(1-z_{3})-1)=\frac{1}{2\pi}d\log z_{1}\wedge d\log z_{2}\wedge d\log z_{3}.$$

The reader is advised to compare this discussion to Section 1.1.6.