We describe the standard construction how to obtain the family of Calabi-Yau hypersurfaces from our input data [Bat94]. Recall that $\lambda\in\mathbb{Z}^{\Delta\cap(\mathbb{Z}^{d})^{*}}$ and $\nu\in\mathbb{Z}^{\Delta^{\vee}\cap\mathbb{Z}^{d}}$ induce central triangulations $\{0\}\ast S$ of $\Delta$ and $\{0\}\ast T$ of $\Delta^{\vee}$. That is, $\lambda$ and $\nu$ lie in the interior of the secondary cone of the respective triangulation [GKZ94].

We use $\lambda$ to define a (complex) one-parameter family $H_{s}^{\operatorname{af{}f}}$ of affine hypersurfaces in the complex torus $(\mathbb{C}\backslash\{0\})^{d}$, and we use $\nu$ in order to compactify the torus and the hypersurfaces in a projective toric variety. The affine hypersurfaces are given by

$$H_{s}^{\operatorname{af{}f}}:=\{x\in(\mathbb{C}\backslash\{0\})^{d}\ :\ \sum_{m\in\Delta\cap(\mathbb{Z}^{d})^{*}}a_{m}s^{\lambda(m)}x^{m}=0\}$$

The triangulation $\{0\}\ast T$ of $\Delta^{\vee}$ induced by the function $\nu:\Delta^{\vee}\cap\mathbb{Z}\rightarrow\mathbb{Z}$ defines a simplicial subdivision of the normal fan to $\Delta$, which in turn defines a projective toric variety $X_{\Delta_{\nu}}$ that contains $(\mathbb{C}\backslash\{0\})^{d}$ as a dense open subset [Ful93, Oda88, Dan78]. Let $H_{s}$ be the closure of $H_{s}^{\operatorname{af{}f}}$ in $X_{{\Delta_{\nu}}}$.

The function $\nu$ determines the class of an ample line bundle on $X_{{\Delta_{\nu}}}$, hence a Kähler class $[\nu]\in H^{2}(X_{{\Delta_{\nu}}},\mathbb{Z})$. There are several, more or less canonical, ways to define a $\mathbb{T}$-invariant Kähler form on $X_{{\Delta_{\nu}}}$ in the class $[\nu]$. One of the possible constructions of a toric variety is via symplectic reduction. In this case $X_{\Delta_{\nu}}$ inherits the natural symplectic structure (which is, in fact, Kähler) from the standard Kähler form $\frac{\sqrt{-1}}{4\pi}\sum dz\wedge d\bar{z}$ on $\mathbb{C}^{\operatorname{vert}(T)}$ (cf. [Gui94]). Alternatively, we can use the pullback of the Fubini-Study form $\omega_{\mathrm{F}S}$ from a projective embedding of $X_{\Delta_{\nu}}$. Though ${\Delta_{\nu}}$ is not necessarily an integral polytope, some $k$-multiple of it certainly is. We can use the complete linear system given by $k{\Delta_{\nu}}$ to define the embedding $i:X_{\Delta_{\nu}}\hookrightarrow\mathbb{C}\mathbb{P}^{(k{\Delta_{\nu}})\cap(\mathbb{Z}^{d})^{*}-1}.$ The Kähler form in the class $[\nu]\in H^{2}(X_{\Delta_{\nu}},\mathbb{Z})$ is then given by $\frac{1}{k}i^{*}\omega_{\mathrm{F}S}$.

In a sense any such “canonical” form $\omega_{0}$ is unsatisfactory because the metric on $H_{s}$ defined by restriction of $\omega_{0}$ to $H_{s}$ is too far from being Ricci-flat. In the second part of this paper we will describe a family of forms $\omega_{s}$ on $X_{\Delta_{\nu}}$, such that the induced metrics on $H_{s}$ approximate the Calabi-Yau metrics as $s\to\infty$ much better. (See the Outlook section for more details).

According to [GKZ94, Ch. 10], the hypersurfaces given by equations in the form $\sum b_{m}x^{m}=0$ are all diffeomorphic to each other (in the orbifold sense) as long as the vector $(\log|b_{m}|)$ ($=\lambda\cdot\log|s|+\log|a|$ in our case) lies in some parallel translation of the $(S\ast\{0\})$-secondary cone. So that the properties of the family related to the smooth structure do not depend on along which ray we approach the large complex structure point ($s\to\infty$). Thus, any vector $\lambda$ in this secondary cone will determine the diffeomorphic torus fibration. For the same reason we can set the coefficients $a_{m}=1$ in the defining equation without loss of generality. On the other side, any choice of the Kähler class, as long as it is in the right Kähler cone, also gives rise to the same combinatorics.

The goal of the rest of this section is to exhibit a torus fibration $H_{s}^{\mathrm{sm}}\to\Sigma\backslash N(D)$ on a “smooth” part of $H_{s}$, for large enough $s$, and show that it is the same as our model fibration $W^{\epsilon}\to\Sigma\backslash N(D)$.