The $S^{1}$-action can be identified as in Proposition 2.11. The holomorphic volume form is characterised by $\iota_{\frac{\partial}{\partial\theta}}\Omega^{(1)}=d\eta_{1}\wedge d\eta_{2}$, which is compared to

to yield $\Omega^{(1)}=-\frac{\sqrt{-1}}{4\pi^{2}z_{1}z_{2}}dz_{2}\wedge dz_{3}\wedge dz_{4}$.

This holomorphic volume form formula in particular shows the map $M^{-}\to\{z_{3}z_{4}=1-z_{1}-z_{2}\}$ is a local biholomorphism wherever the complex structure is defined. We finally need to show this map is injective. Since both $M^{-}$ and $\{z_{3}z_{4}=1-z_{1}-z_{2}\}$ fibre over $\mathbb{C}^{*}_{z_{1}}\times\mathbb{C}^{*}_{z_{2}}$ in a compatible way, it suffices to compare the $\mathbb{C}^{*}$-fibres. The map between the fibres is equivariant with respect to the $S^{1}$-action, so to conclude injectivity we only need to recall from the proof of Lemma 4.28 that $\log|z_{3}|$ is a monotone function of $\mu$. ∎