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 $\eta$. ∎