By Lemma 2.14 the initial volume form error is

$$\left\lVert E^{(2)}\right\rVert_{C^{k,\alpha}_{-1-\epsilon,-1+\epsilon}}\leq\left\lVert E^{(2)}\right\rVert_{C^{k,\alpha}_{-1,-1}}\leq C.$$

Applying Corollary 2.24 we can solve the Poisson equation with estimate

$$\Delta_{g^{(2)}}u_{1}=-2E^{(2)},\quad\left\lVert du_{1}\right\rVert_{C^{k+1,\alpha}_{-\epsilon,-1+\epsilon}(\mathbb{C}^{3},\Lambda^{1})}\leq CA^{-1/4},$$

so in particular

$$\left\lVert\partial\bar{\partial}u_{1}\right\rVert_{C^{k,\alpha}_{-1-\epsilon,-1+\epsilon}}\leq C,\quad\left\lVert(\partial\bar{\partial}u_{1})^{2}\right\rVert_{C^{k,\alpha}_{-2-2\epsilon,-2+2\epsilon}}\leq C.$$

Now $(\omega^{(2)^{\prime}})^{3}=(\omega^{(2)}+\sqrt{-1}\partial\bar{\partial}u_{1})^{3}=(\omega^{(2)})^{3}(1+\frac{1}{2}\Delta_{g^{(2)}}u_{1}+O(|\partial\bar{\partial}u_{1}|^{2}))$, so the new volume form error has improved decay:

$$\frac{3}{4}(E^{(2)^{\prime}}+1)\sqrt{-1}\Omega\wedge\overline{\Omega}=(\omega^{(2)^{\prime}})^{3},\quad\left\lVert E^{(2)^{\prime}}\right\rVert_{C^{k,\alpha}_{-2,-2+2\epsilon}}\leq\left\lVert E^{(2)^{\prime}}\right\rVert_{C^{k,\alpha}_{-2-2\epsilon,-2+2\epsilon}}\leq C.$$

We notice that the modification to $\omega^{(2)}$ is $C^{0}$-small outside a compact region, where the positive definite condition for the Kähler metric is not affected. Inside the compact set we can add on a locally supported semipositive (1,1)-form to guarantee the Kähler condition, as we have done in Section 2.6. We abuse notation to write this Kähler metric after surgery as $\omega^{(2)^{\prime}}$, which inherits all the analytic properties of $\omega^{(2)}$.

Applying Corollary 2.24 again to solve the Poisson equation with background metric $g^{(2)^{\prime}}$,

$$\Delta_{g^{(2)^{\prime}}}u_{2}=-2E^{(2)^{\prime}},\quad\left\lVert du_{2}\right\rVert_{C^{k+1,\alpha}_{-1,-2+2\epsilon}(\mathbb{C}^{3},\Lambda^{1})}\leq CA^{-1/4},$$

and using $(\omega^{(2)^{\prime}}+\sqrt{-1}\partial\bar{\partial}u_{2})^{3}=(\omega^{(2)^{\prime}})^{3}(1+\frac{1}{2}\Delta_{g^{(2)^{\prime}}}u_{2}+O(|\partial\bar{\partial}u_{2}|^{2})),$ the new volume form error is now bounded in $C^{k,\alpha}_{-4,-4+4\epsilon}$-norm. Another surgery in the compact region ensures the Kähler property. ∎

We assume $C^{-1}\delta_{ij}\leq a_{ij}\leq C\delta_{ij}$ which can be relaxed by scaling. It suffices to solve the complex Monge-Ampère equation

$$(\omega^{(3)}+\sqrt{-1}\partial\bar{\partial}\phi^{\prime})^{3}=\frac{3}{4}\sqrt{-1}\Omega\wedge\overline{\Omega}.$$

In Hein’s analytic package (cf. Section 2.7), the conditions on the ambient metric including $C^{k,\alpha}$ quasi-atlas, existence of a distance-like function with Hessian bounds, and the weighted Sobolev inequality, are robust conditions which are inherited by $\omega^{(3)}$ from $\omega^{(2)}$. The volume form error $E^{(3)}$ has faster than quadratic decay by construction:

$$|E^{(3)}|\leq C(|\vec{\mu}|_{a}+1)^{-4+\epsilon}.$$

Thus Hein’s package provides a potential $\phi^{\prime}$ solving the complex Monge-Ampère equation with decay estimate $|\phi^{\prime}|\leq C(|\vec{\mu}|_{a}+1)^{-2+2\epsilon}$. Elliptic bootstrap gives the bound $\left\lVert\phi^{\prime}\right\rVert_{C^{k+2,\alpha}_{0,-2+2\epsilon}}\leq C,$ so $\left\lVert d\phi^{\prime}\right\rVert_{C^{k+1,\alpha}_{-1,-2+2\epsilon}(\mathbb{C}^{3},\Lambda^{1})}\leq C$, which combined with Lemma 2.25 implies the metric deviation estimate. ∎

Using the metric deviation estimate, the Riemannian curvature is bounded. Morever if $\ell>2A^{-1/4}$, then we can find a flat model $g_{\text{flat}}$ over a $g_{a}$-ball of radius $\sim\ell/10$, where $\left\lVert g_{\text{flat}}-g_{\mathbb{C}^{3}}\right\rVert_{C^{k,\alpha}_{-1,0}}\leq C.$ Using this bound up to second order derivatives, the Christoffel symbols in the local flat coordinates are $O(A^{-1/4}\ell^{-2})$ and the Riemannian curvature is of order $O(A^{-1/4}\ell^{-3})$. ∎

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. ∎