The negative vertex $M^{-}$ describes an open subset inside $\{z_{3}z_{4}=1-z_{1}-z_{2}\}\subset\mathbb{C}^{*}_{z_{1}}\times\mathbb{C}^{*}_{z_{2}}\times\mathbb{C}^{2}_{z_{3},z_{4}}$. In the toric hypersurface picture, we are looking at a region where the dominating monomials are up to scale factors just $(z_{3}z_{4})^{-1}$, $z_{1}(z_{3}z_{4})^{-1}$, $z_{2}(z_{3}z_{4})^{-1}$ and 1, so the defining equation of $X_{t}$ is approximately $(1-z_{1}-z_{2})(z_{3}z_{4})^{-1}-1=0$ once we absorb the scale factors into $z_{i}$. The holomorphic volume form $\Omega$ is given up to constant by

$$\frac{\sqrt{-1}}{4\pi^{2}}d\log z_{1}\wedge d\log z_{2}\wedge d\log z_{3}\wedge d\log z_{4}=d\left((1-z_{1}-z_{2})(z_{3}z_{4})^{-1}-1\right)\wedge\Omega,$$

or equivalently $\Omega=\frac{-\sqrt{-1}}{4\pi^{2}}\frac{1}{z_{1}z_{2}}dz_{2}\wedge dz_{3}\wedge dz_{4}.$ This model has $S^{1}$-symmetry:

$$e^{i\theta}\cdot(z_{1},z_{2},z_{3},z_{4})=(z_{1},z_{2},e^{i\theta}z_{3},e^{-i\theta}z_{4}).$$

Hence $M^{-}$ is a singular $S^{1}$-bundle over $\mathbb{R}_{\mu}\times\mathbb{C}^{*}_{z_{1}}\times\mathbb{C}^{*}_{z_{2}}$, where $\mu$ is the $S^{1}$-moment coordinate which takes the value zero on the singular locus $\{z_{3}=z_{4}=0\}$ (notice that the degeneracy of the $S^{1}$ factor implies that the moment map is constant on this singular locus for any choice of Kähler form). This agrees with the modified topological description in Section 1.1.5. We calculate

$$\iota_{\frac{\partial}{\partial\theta}}\Omega=-\frac{1}{4\pi^{2}}d\log z_{1}\wedge d\log z_{2}=d\eta_{1}\wedge d\eta_{2},$$

where the logarithmic coordinates $\eta_{p}=\frac{1}{2\pi\sqrt{-1}}\log z_{p}$ for $p=1,2$ have period 1. The $\arg z_{1},\arg z_{2}$ coordinates provide a family of 2-tori in $\mathbb{R}\times\mathbb{C}^{*}_{z_{1}}\times\mathbb{C}^{*}_{z_{2}}$, and the restriction of the $S^{1}$-bundle over these 2-tori defines a family of 3-tori. The normalising constant on $\Omega$ imply that $\int_{T^{3}}\Omega=2\pi$.