The existence of moment coordinates follows from $H^{1}(\mathbb{C}^{3})=0$, but for the purpose of estimation we wish to relate $\tilde{\mu}_{i}^{\mathbb{C}^{3}}$ to $\mu_{i}$ outside the ball $\{|\vec{\mu}|_{a}\lesssim A^{-1/4}\}$ where the surgery was performed. In this exterior region

$$\omega_{\mathbb{C}^{3}}=\omega^{(2)}+\sqrt{-1}\partial\bar{\partial}\phi^{\mathbb{C}^{3}}=\omega^{(1)}+dd^{c}\phi^{\mathbb{C}^{3}},\quad d^{c}=\frac{\sqrt{-1}}{2}(\bar{\partial}-\partial).$$

The 1-form $d^{c}\phi^{\mathbb{C}^{3}}$ is $T^{2}$-invariant, so by Cartan’s formula

$$\iota_{\frac{\partial}{\partial\theta_{i}}}dd^{c}\phi^{\mathbb{C}^{3}}=-d\iota_{\frac{\partial}{\partial\theta_{i}}}d^{c}\phi^{\mathbb{C}^{3}},$$

which combined with $d\mu_{i}=-\iota_{\frac{\partial}{\partial\theta_{i}}}\omega^{(1)}$ allow us to find the moment coordinates:

$$\tilde{\mu}_{i}^{\mathbb{C}^{3}}=\mu_{i}+\iota_{\frac{\partial}{\partial\theta_{i}}}d^{c}\phi^{\mathbb{C}^{3}},\quad d\tilde{\mu}_{i}^{\mathbb{C}^{3}}=-\iota_{\frac{\partial}{\partial\theta_{i}}}\omega_{\mathbb{C}^{3}},\quad i=1,2.$$

Using the estimates $|d\phi^{\mathbb{C}^{3}}|\leq CA^{-1/2}\ell^{-\epsilon}|\vec{\mu}|_{a}^{-1+\epsilon}$ and $|\frac{\partial}{\partial\theta_{i}}|\leq CA^{-1/4},$ we see

$$|\mu_{i}-\tilde{\mu}_{i}^{\mathbb{C}^{3}}|\leq CA^{-3/4}\ell^{-\epsilon}|\vec{\mu}|_{a}^{-1+\epsilon}.$$

Now inside $\{|\vec{\mu}|_{a}\lesssim A^{-1/4}\}$, we have $|\mu_{i}|\leq CA^{-1/2}$, and $|d\tilde{\mu}_{i}|\leq CA^{-1/4}$ integrates to give $|\tilde{\mu}_{i}|\leq CA^{-1/2}$. Thus globally on $\mathbb{C}^{3}$

$$|\mu_{i}-\tilde{\mu}_{i}^{\mathbb{C}^{3}}|\leq CA^{-3/4}\ell^{-\epsilon}(A^{-1/4}+|\vec{\mu}|_{a})^{-1+\epsilon}$$

as required. Morever $\tilde{\mu}_{1}^{\mathbb{C}^{3}},\tilde{\mu}_{2}^{\mathbb{C}^{3}},\tilde{\mu}_{1}^{\mathbb{C}^{3}}-\tilde{\mu}_{2}^{\mathbb{C}^{3}}$ vanish respectively along $\mathfrak{D}_{1},\mathfrak{D}_{2},\mathfrak{D}_{3}$, due to the respective vanishing of the circle generators $\frac{\partial}{\partial\theta_{1}},\frac{\partial}{\partial\theta_{2}},\frac{\partial}{\partial\theta_{1}}-\frac{\partial}{\partial\theta_{2}}$.

Now consider the map $\mathbb{C}^{3}\xrightarrow{(\tilde{\mu}_{1}^{\mathbb{C}^{3}},\tilde{\mu}_{2}^{\mathbb{C}^{3}},\text{Im}(\eta))}\mathbb{R}^{3}$. It is a special Lagrangian fibration by Remark 1.6.

At a critical point $p\in\mathbb{C}^{3}$ the Zariski tangent space of the fibre, namely the annihilator of $\text{span}(d\tilde{\mu}_{1}^{\mathbb{C}^{3}},d\tilde{\mu}_{2}^{\mathbb{C}^{3}},d\text{Im}\eta)$, is a linear subspace of $T_{p}\mathbb{C}^{3}$ of real dimension at least 4. It contains $\frac{\partial}{\partial\theta_{1}},\frac{\partial}{\partial\theta_{2}}$ and is $g_{\mathbb{C}^{3}}$-orthogonal to $I\frac{\partial}{\partial\theta_{1}},I\frac{\partial}{\partial\theta_{2}}$. If $d\eta\neq 0$ at $p$, then $\frac{\partial}{\partial\theta_{1}},\frac{\partial}{\partial\theta_{2}}$ are linearly independent, so the Zariski tangent space is the orthogonal complement of $\text{span}(I\frac{\partial}{\partial\theta_{1}},I\frac{\partial}{\partial\theta_{2}})$ by dimension counting. Since $d\text{Im}\eta$ vanishes on the Zariski tangent space, and $d\eta$ vanishes on $\text{span}_{\mathbb{C}}(\frac{\partial}{\partial\theta_{1}},\frac{\partial}{\partial\theta_{2}})$, we deduce $d\eta=0$ on $T_{p}\mathbb{C}^{3}$, contradiction. Thus the critical points must satisfy $d\eta=0$, or equivalently $p\in\bigcup_{i,j}\{z_{i}=z_{j}=0\}$. Conversely all points in $\bigcup_{i,j}\{z_{i}=z_{j}=0\}$ are critical. Having identified the critical point set, the discriminant locus follows from the argument in Lemma 1.6. ∎