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

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 $W^{\epsilon}\to\Sigma\backslash N(D)$.

Let $\mathrm{Log}_{s}:(\mathbb{C}\backslash\{0\})^{d}\rightarrow\mathbb{R}^{d}$ be the logarithmic map with the base $|s|$:

The geometry of amoebas of affine hypersurfaces is a well developed subject that originated in the work of Gelfand, Kapranov and Zelevinsky [GKZ94]. We are going to review several useful facts about the amoebas, most of which are contained in (or can be easily deduced from) a nice survey paper by Mikhalkin [Mik01].

The limiting behavior of amoebas as $s\to\infty$ can be described in terms of the Legendre transform $L_{\lambda}:\mathbb{R}^{d}\rightarrow\mathbb{R}$ of the vector $\lambda$:

$L_{\lambda}(n)$ is a piecewise linear convex function. Define the non-Archimedean amoeba $\mathcal{A}^{\lambda}_{\infty}\subset\mathbb{R}^{d}$ to be the corner locus of $L_{\lambda}(n)$ (the set of points where $L_{\lambda}(n)$ is not smooth). $\mathcal{A}^{\lambda}_{\infty}$ is a rational polyhedral complex of dimension $d-1$ (cf. [Mik02]), which gives a cell decomposition of $\mathbb{R}^{d}$.

The polyhedral complex $\mathcal{A}^{\lambda}_{\infty}$ is also called the spine of the amoeba $\mathcal{A}^{\lambda}_{s}$ because of the following fact (cf. [Mik02]):

Given a $\mathbb{T}$-invariant Kähler form on $X_{{\Delta_{\nu}}}$ in the class $[\nu]$, we can consider the corresponding moment map $\mu\colon X_{{\Delta_{\nu}}}\to{{\Delta_{\nu}}}$. In this case we can also define the compactified amoeba $\overline{\mathcal{A}^{\lambda}_{s}}:=\mu(H_{s})\subset{\Delta_{\nu}}$.

For the proof of Theorem 3.7 we will need to introduce some domains in $\mathbb{R}^{d}$, which are intimately connected with the amoebas and the function $L_{\lambda}$. In some sense they are generalizations of the cells induced by $\mathcal{A}^{\lambda}_{\infty}$.

Notice that for small $\epsilon$, $Q^{\lambda}_{(I\mid J)}(\epsilon)$ and $Q^{\lambda}_{(I\mid J)}(0)$ are combinatorially equivalent.

We will exhibit a vector field on $\mathbb{R}^{d}\backslash Q^{\lambda}_{\{0\}}(\epsilon)$ with values in $\partial\Delta^{\vee}_{\lambda}\subset\mathbb{R}^{d}$. Its integral curves yield the desired foliation $\mathcal{F}$. Recall from Lemma 2.3 that there is a subdivision of $\partial\Delta^{\vee}_{\lambda}$ which is combinatorially isomorphic to $\operatorname{bsd}(S)\times T$, restricted to $|\Sigma|$. In this context, the projection $p_{2}$ can be thought of as defined $\partial\Delta^{\vee}_{\lambda}\rightarrow{\partial\Delta^{\vee}}$, and can be interpreted as a vector field on $\partial\Delta^{\vee}_{\lambda}$. We will deform $p_{2}$, and extend the deformed vector field to $\mathbb{R}^{d}\backslash Q^{\lambda}_{\{0\}}(\epsilon)$.

For the deformation part, we use that the faces of $\operatorname{bsd}(T,T)$ are parameterized by pairs $(\tau,$ $\mathfrak{T}$ $=\tau_{0}\prec\ldots\prec\tau_{r})\in T\times\operatorname{bsd}(T)$ with $\tau\preceq\tau_{0}$ (cf. § 2.2).

We will apply Lemma 3.5 for $K=p_{2}(\partial\mathcal{V})$, and $N=p_{2}(N_{2}(\partial\mathcal{V}))$, where $N_{2}(\partial\mathcal{V})$ is a neighborhood of $\partial\mathcal{V}$ in $\Sigma$.

Now we are ready to define the vector field $\mathfrak{X}^{\delta}$ on $\partial\Delta^{\vee}_{\lambda}$ as the composition $\mathfrak{d}^{\delta}p_{2}\colon\partial\Delta^{\vee}_{\lambda}\rightarrow{\partial\Delta^{\vee}}$. In order to extend $\mathfrak{X}^{\delta}$ to both sides of $\partial\Delta^{\vee}_{\lambda}$, we present a polyhedral subdivision of a neighborhood of $\partial\Delta^{\vee}_{\lambda}$ whose trace on $\partial\Delta^{\vee}_{\lambda}$ realizes the restriction of $\operatorname{bsd}(S)\times T$ to $|\Sigma|$.

Denote by $\lambda^{\epsilon}\in\mathbb{R}^{\Delta\cap(\mathbb{Z}^{d})^{*}}$ the vector given by $\lambda^{\epsilon}(0)=\lambda(0)$, and $\lambda^{\epsilon}(v)=\lambda(v)+\epsilon$ for $v\in\operatorname{vert}(S)$. Suppose that $\epsilon>0$ is small enough to ensure that $\lambda$ and $\lambda^{\epsilon}$ induce the same triangulation. Then $\Delta^{\vee}_{\lambda^{\epsilon}}\subset\Delta^{\vee}_{\lambda}$ are combinatorially equivalent. For a simplex $\mathfrak{S}$ $\in\operatorname{bsd}(S)$, denote $\mathfrak{S}$_ϵ the corresponding simplex of $\operatorname{bsd}(\Delta^{\vee}_{\lambda^{\epsilon}})$. (I.e., $\mathfrak{S}$ $\subset(\mathbb{R}^{d})^{*}$, while $\mathfrak{S}$${}_{\epsilon}\subset\mathbb{R}^{d}$.) Given a simplex $\tau\in T$ with $\langle$$\mathfrak{S}$$,\tau\rangle=1$, we can form the Minkowski sum $\mathfrak{S}$${}_{\epsilon}+\mathbb{R}_{\geq\epsilon/2}\tau$. These fit together to form a complex of (unbounded) polyhedra which subdivides $\mathbb{R}^{d}$ outside $\Delta^{\vee}_{\lambda^{\epsilon/2}}$. (It actually refines the subdivision into $Q^{\lambda^{\epsilon/2}}_{\sigma}$’s provided by the non-Archimedean amoeba for $\lambda^{\epsilon/2}$.)

We need to argue that this determines a continuous vector field. The problem may arise only when we try to assign a vector to a point $n\in$ $\mathfrak{S}$${}_{\epsilon}+\mathbb{R}_{\geq\epsilon/2}\tau$ such that there are two points $n_{1},n_{2}$ which already have a vector assigned to them, so that both $n-n_{1}$ and $n-n_{2}$ are a multiple of $\widehat{\tau}$. Then $n_{1}\in$ $\mathfrak{S}$${}_{\epsilon}+[\epsilon/2,\epsilon]\tau_{1}$ for some face $\tau_{1}\prec\tau$, and $\mathfrak{X}^{\delta}(n_{1})$ is defined as $\mathfrak{X}^{\delta}(n_{1}^{\prime})$, where $n_{1}^{\prime}\in\partial\Delta^{\vee}_{\lambda}$, and $n_{1}-n_{1}^{\prime}$ is a multiple of $\widehat{\tau}_{1}$. Furthermore, $n_{2}\in\partial\Delta^{\vee}_{\lambda}$, and $n_{2}-n_{1}^{\prime}$ is a multiple of $\widehat{\tau}-\widehat{\tau}_{1}$. Here we use the assumption that $\delta<1/2$ to conclude that $\mathfrak{X}^{\delta}(n_{2})=\mathfrak{d}^{\delta}p_{2}(n_{2})=\mathfrak{d}^{\delta}p_{2}(n_{1}^{\prime})=\mathfrak{X}^{\delta}(n_{1}^{\prime})$.

The integral curves of $\mathfrak{X}^{\delta}$ foliate $\mathbb{R}^{d}\backslash\Delta^{\vee}_{\lambda^{\epsilon/2}}$. The following lemma summarizes the main properties of $\mathfrak{X}^{\delta}$ and the foliation $\mathcal{F}$.

Using the foliation $\mathcal{F}$ we are going to define a decomposition of the hypersurface $H_{s}=H_{s}^{\mathrm{sm}}\sqcup H_{s}^{\mathrm{sing}}$, construct a torus fibration $H_{s}^{\mathrm{sm}}\rightarrow\Sigma\backslash N(D)$ and show that it is isomorphic to the fibration $W^{\epsilon}\rightarrow\Sigma\backslash N(D)$.

For any closed subset $J\subset\Sigma$ we will denote by $X_{s}(J)\subset X_{\Delta_{\nu}}$ the closure of $\mathrm{Log}_{s}^{-1}\left(\bigcup_{q\in J}\mathcal{F}_{q}\right)$ in $X_{\Delta_{\nu}}$.

Since $D=\partial\mathcal{U}\cap\partial\mathcal{V}$, there exist regular neighborhoods $N_{1}(\partial\mathcal{U})$ of $\partial\mathcal{U}$ and $N_{2}(\partial\mathcal{V})$ of $\partial\mathcal{V}$ in $\Sigma$, such that $N(D)\supset N_{1}(\partial\mathcal{U})\cap N_{2}(\partial\mathcal{V})$. This means that $\Sigma\backslash N(D)$ can be covered by the union of the closed sets:

The amoebas $\mathcal{A}^{\lambda}_{s}$, for a large enough $s$, all lie in $\mathbb{R}^{d}\backslash Q^{\lambda}_{\{0\}}(\epsilon)$. This means that $\mathcal{F}$ defines a projection $\mathcal{A}^{\lambda}_{s}\to\Sigma$ and, by composition with $\mathrm{Log}_{s}$, the projection $H_{s}^{\operatorname{af{}f}}\to\Sigma$. Also $\mathcal{A}^{\lambda}_{s}$ lie in $\mathbb{R}^{d}\backslash Q^{\lambda}_{v}(\epsilon)$, for any $v\in\operatorname{vert}(S)$ and large $s$. Since the unbounded ends of flow lines $\mathcal{F}_{q}$, for $q\in U^{\epsilon}_{v}$, are in $Q^{\lambda}_{v}(\epsilon)$ their closures do not contain any extra points of the hypersurface:

Thus, the map $H_{s}\cap X_{s}(U^{\epsilon}_{v})\to U^{\epsilon}_{v}$ is well defined. On the other hand, for two distinct points $q_{1},q_{2}$ in $V^{\delta}_{w}$ the corresponding leaves are straight lines. Written in local coordinates (see Lemma 3.9) this implies that the sets $X_{s}(\mathcal{F}_{q_{1}})$ and $X_{s}(\mathcal{F}_{q_{2}})$ are disjoint. Hence, the map $H_{s}\cap X_{s}(V^{\delta}_{w})\to V^{\delta}_{w}$ is well defined. Combined together we have (for large enough $s$) the well defined projection

Figure 16: The foliation $\mathcal{F}$ of $\mathbb{R}^{d}\backslash Q^{\lambda}_{\{0\}}(\epsilon)$ induces a foliation of the amoeba $\mathcal{A}^{\lambda}_{s}$.

Before proving the theorem we need to make a comment about smoothness. $H_{s}^{\mathrm{sm}}$ is missing all singular points (if any) of the toric variety $X_{{\Delta_{\nu}}}$, which are all in the moment map preimage of the $(d-2)$-skeleton of ${\Delta_{\nu}}$ (see Lemma 3.9). On the other hand, $\Sigma\backslash N(D)$ carries a canonical smooth structure induced by the affine structure on $\Sigma\backslash D$. So given the topological fibration $H_{s}^{\mathrm{sm}}\rightarrow\Sigma\backslash N(D)$ of Theorem 3.7, standard techniques apply to make it smooth.

The strategy of proving Theorem 3.7 will be as follows. First, we analyze the map $f_{s}$ in $U^{\epsilon}_{v}$ and $V^{\delta}_{w}$ for every $v\in\operatorname{vert}(S),w\in\operatorname{vert}(T)$. Then the proof of the theorem can be completed in three steps: we show that $f_{s}\colon H_{s}^{\mathrm{sm}}\to\Sigma\backslash N(D)$ is a torus fibration over the two kinds of covering patches, and then check that it has the monodromy of our model.

Let $v\in\operatorname{vert}(S)$. For a fixed $s$ we consider the $(\Delta\cap(\mathbb{Z}^{d})^{*}-2)$-parameter family of hypersurfaces $H_{s}^{v}(a)$ in $X_{s}(U^{\epsilon}_{v})$: