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

$$-\frac{\sqrt{-1}}{2\pi}d\log z_{0}\wedge d\log z_{1}\wedge d\log z_{2}\wedge d\log z_{3}=d((z_{0}z_{1}z_{2})^{-1}(1-z_{3})-1)\wedge\Omega,$$

or equivalently $\Omega=-\frac{\sqrt{-1}}{2\pi}\frac{1}{z_{3}}dz_{0}\wedge dz_{1}\wedge dz_{2}$. An important feature of this model is the diagonal $T^{2}$-symmetry:

$$e^{i\theta_{1}}\cdot(z_{0},z_{1},z_{2})=(e^{-i\theta_{1}}z_{0},e^{i\theta_{1}}z_{1},z_{2}),\quad e^{i\theta_{2}}\cdot(z_{0},z_{1},z_{2})=(e^{-i\theta_{2}}z_{0},z_{1},e^{i\theta_{2}}z_{2}).$$

We have $\Omega(\frac{\partial}{\partial\theta_{1}},\frac{\partial}{\partial\theta_{2}},\cdot)=-\frac{\sqrt{-1}}{2\pi}d\log z_{3}=d\eta$, where $\eta=-\frac{\sqrt{-1}}{2\pi}\log(z_{3})$ is a holomorphic coordinate with period 1, and takes the value zero at $z_{0}=z_{1}=z_{2}=0$. The relation between this complex geometric perspective and the topological picture in Section 1.1.3 is perhaps clearest with the generalised Gibbons-Hawking construction in mind (cf. Section 1.2 below). Essentially $M^{+}$ is a singular $T^{2}$-bundle over a 4-dimensional base contained in $\mathbb{R}^{2}_{\mu_{1},\mu_{2}}\times(S^{1}\times\mathbb{R})_{\eta}$, where $\mu_{1},\mu_{2}$ are the $T^{2}$-moment maps normalised to have value 0 at $z_{0}=z_{1}=z_{2}=0$. We shall notice that the discriminant locus $\mathfrak{D}\times\{0\}$ of this singular $T^{2}$-bundle is not sensitive to the choice of the Kähler form (cf. Lemma 1.6 and its ensuing Remark). The normalising constant on $\Omega$ imply that the SYZ $T^{3}$-fibres have $\int_{T^{3}}\Omega=4\pi^{2}$.