In [HZ02] we have used the GKZ machinery [GKZ94] for the monomial estimates in the equation of $Z_{a}$ to establish an embedding of $Z_{a}^{\mathrm{sm}}$ into $W_{a}:=W(\log|a|,\operatorname{Arg}(a))$. The same estimates can be used to find bounds on the discrepancy of this embedding from being holomorphic and isometric. Establishing these bounds will occupy the rest of the section.

To define the fibration we assume that $\lambda=\log|a|$ is sufficiently far in the interior of $\operatorname{SC}(S)$, so that $c:=\log|\Delta_{\mathbb{Z}}|$ is small in the $\lambda$-scale.

From now on we will use the $(\lambda-c,\nu)$ bi-PIKAS to identify $\Sigma$ with $\partial\Delta^{\vee}_{\lambda-c}$ and fix the vector field and the foliation in $\mathbb{R}^{d}\backslash\Delta^{\vee}_{\lambda-c}$. With this identification, given a subset $U\in\Sigma$ we denote by $X(U)$ the closure of the set $\log^{-1}(\cup_{q\in U}\mathcal{F}_{q})$ in the toric variety $X_{T}$ (cf. [HZ02]). Then the smooth part $Z_{a}^{\mathrm{sm}}$ of the hypersurface is defined as:

We define the map $\psi$ over the charts $V_{w}^{\beta^{\vee}}$ as the restriction to $Z_{a}$ of the quotient map:

Note that if $x$ is in a boundary toric divisor $Z_{w},w\in\operatorname{vert}(T)$, then $\log|z|$ and $\operatorname{Arg}(z)$ are not well defined, but $\log|z|/w$ and $\operatorname{Arg}(z)/w$ are. Hence, the map $\psi$ is well defined over $X(V_{w}^{\beta^{\vee}})$. The meaning of this map is the choice of local coordinates for $Z_{a}$ near $z$ (cf. [HZ02, Lemma 3.9]). Hence it is holomorphic.

Before defining the map over the charts $U_{v}^{\beta}$ let us first make some estimates in the spirit of Lemma 4.1. For an element $\theta\in\mathbb{R}/\mathbb{Z}$ we will write $|\theta|<c$ if the (standard Euclidean) distance from $\theta$ to 0 is less than $c$. This inequality is vacuous for $c\geq 1/2$.

Now, identifying the tangent spaces of the torus fibers $\mathbb{T}_{x}=\log^{-1}(x)$ with $\mathbb{R}^{d}$, we can pull back the vector field $\mathfrak{X}(x)$ to get a (constant) vector field on $\mathbb{T}_{x}$. Then the map $\psi$ for the points in $X(U_{v}^{\beta})\cap Z_{a}\subset(\mathbb{C}\backslash\{0\})^{d}$ will be defined as:

Here if $\beta$ is large enough, i.e. $C(\beta)e^{-\beta}<\frac{1}{2}$, then according to the Lemma 4.2 there is a preferred continuous lift of $\langle v,\operatorname{Arg}(z)\rangle+\theta_{v}$ to (the neighborhood of 0 in) $\mathbb{R}$. We use this lift to first define the value for $(\langle v,\operatorname{Arg}(z)\rangle+\theta_{v})\mathfrak{X}$ in $\mathbb{R}^{d}$ and then project it back to $\mathbb{T}=\mathbb{R}^{d}/\mathbb{Z}^{d}$.

Since $\mathfrak{X}_{q}=w$ for $q\in V_{w}^{\beta^{\vee}}$ the definition of the map $\psi$ over the charts $V_{w}^{\beta^{\vee}}$ is consistent with the above definition on possible overlaps $U_{v}^{\beta}\cap V_{w}^{\beta^{\vee}}$. Thus, we have a well defined map $\psi:Z_{a}^{\mathrm{sm}}\to W_{a}$ which is an embedding [HZ02].