By similar calculations as in Corollary 2.29, the moment map on $M^{+}_{\nu}$ is expressed as

where $\phi_{4}$ is the Kähler potential between $\tilde{\omega}^{(4)}$ and $\tilde{\omega}^{(3)}$ (cf. Section Section 3.7), and $\tilde{\mu}_{i}^{\mathbb{C}^{3}}$ are the moment coordinates for $\omega_{\mathbb{C}^{3}}$ (cf. Corollary 2.29). By construction $\tilde{\mu}_{1},\tilde{\mu}_{2},\tilde{\mu}_{1}-\tilde{\mu}_{2}$ vanish respectively along $\mathfrak{D}_{1},\mathfrak{D}_{2},\mathfrak{D}_{3}$, due to the respective vanishing of the circle generators $\frac{\partial}{\partial\theta_{1}},\frac{\partial}{\partial\theta_{2}},\frac{\partial}{\partial\theta_{1}}-\frac{\partial}{\partial\theta_{2}}$. This fixes the additive normalisation on the moment coordinates.

The gradient estimates on Kähler potentials and Corollary 2.29 imply on $M^{+}_{\nu}$

In particular, if $|(\tilde{\mu}_{1},\tilde{\mu}_{2},y)|_{a}^{\prime}\leq\frac{1}{2}A^{1/2}e^{\nu}$, then $\varrho\leq\frac{3}{4}A^{1/2}e^{\nu}$, so the map (3.14) is proper over $\{|(\tilde{\mu}_{1},\tilde{\mu}_{2},y)|_{a}^{\prime}\leq\frac{1}{2}A^{1/2}e^{\nu}\}\subset\mathbb{R}^{3}$. By the same argument in Corollary 2.29, the fibres of (3.14) are special Lagrangians of phase angle zero, the critical point set is $\bigcup_{i,j\in\{0,1,2\}}\{\tilde{Z}_{i}=\tilde{Z}_{j}=0\}$ and the discriminant locus is contained in $\mathfrak{D}$.

Next we consider the map $(\mu_{1},\mu_{2},\eta)\mapsto(\tilde{\mu}_{1},\tilde{\mu}_{2},\eta)$ on the region $\{\varrho<A^{1/2}e^{\nu}\}$. Using (3.15) and the implicit function theorem, this map restricted to the region $\{\ell\geq A^{1/2},\varrho<\frac{3}{4}A^{1/2}e^{\nu}\}$ is an approximate identity, and in particular a diffeomorphism onto its image. Morever by (3.15) no points elsewhere can map into $\text{Image}(\{\ell\geq A^{1/2},\varrho<\frac{3}{4}A^{1/2}e^{\nu}\})$. Interpreted geometrically, this implies that the special Lagrangian fibres of (3.14) lying over the region $\{|(\tilde{\mu}_{1},\tilde{\mu}_{2},y)|_{a}^{\prime}\leq\frac{1}{2}A^{1/2}e^{\nu}\}$ and suitably away from $\mathfrak{D}$, must be small perturbations of the $T^{3}$-fibres of the map $M^{+}_{\nu}\xrightarrow{(\mu_{1},\mu_{2},y)}\mathbb{R}^{3}$. This shows the generic fibre of (3.14) is topologically $T^{3}$, and the monodromy data of (3.14) is the same as for $M^{+}\xrightarrow{(\mu_{1},\mu_{2},y)}\mathbb{R}^{3}$, which by construction agrees with the Gross-Ruan prediction in Section 1.1.3.

Finally we need to determine the topology of the central singular fibre, defined as the set $X_{0}=\{\tilde{\mu}_{1}=\tilde{\mu}_{2}=0,y=0\}$, which is invariant under the $T^{2}$-action. From our knowledge of the critical point set, the only singular point on the central fibre is $\tilde{Z}_{0}=\tilde{Z}_{1}=\tilde{Z}_{2}=0$. Thus the quotient $X_{0}/T^{2}$ must be a compact 1-dimensional manifold with possibly one singular point. But there is also a homological constraint

so $X_{0}/T^{2}$ is connected and must in fact be a circle. Therefore $X_{0}$ has the topology of $T^{3}$ with a copy of $T^{2}$ collapsed to a point, in accordance with the Gross-Ruan prediction on the positive vertex. ∎