Let $\mathbb{A}^{n}$ be $n$-dimensional affine space. An integral affine structure on an $n$-dimensional manifold $Y$ is given by an open covering $\{U_{\alpha}\}$ of $Y$ together with coordinates $\phi_{\alpha}:U_{\alpha}\to\mathbb{A}^{n}$ such that the transition maps $\phi_{\alpha}\circ\phi_{\beta}^{-1}$ are in $\operatorname{SL}(n,\mathbb{Z})\ltimes\mathbb{R}^{n}$ on the non-empty overlaps $U_{\alpha}\cap U_{\beta}$. An integral Kähler affine structure on $Y$ is a Riemannian metric $g$ which is potential in local affine coordinates, i.e. $g_{ij}=\frac{\partial^{2}K_{\alpha}}{\partial y_{i}\partial y_{j}}$ for some local potentials $K_{\alpha}$.

The dual Kähler affine structure on the same Riemannian manifold $(Y,g)$ is defined as follows (cf [KS01]). We use the same covering $\{U_{\alpha}\}$. The new affine coordinates are $\hat{y}_{i}=\frac{\partial K_{\alpha}}{\partial y_{i}}$ which take values in the dual affine space $({\mathbb{A}^{n}})^{*}$ (the underlying vector spaces for ${\mathbb{A}^{n}}$ and $({\mathbb{A}^{n}})^{*}$ are naturally dual). The new local potentials $\hat{K}_{\alpha}$ are defined by the Legendre transforms of the old ones:

Here one needs to choose origins in ${\mathbb{A}^{n}}$ and $(\mathbb{A}^{n})^{*}$ to define the pairing. Different choices give rise to equivalent Kähler affine structures. The dual affine structure is integral iff the original one is.

Given an affine structure on $Y$ one can consider its monodromy representation $\pi_{1}(Y)\to SL(n,\mathbb{Z})\ltimes\mathbb{R}^{n}$. Two equivalent affine structures have conjugate monodromies.

For a Kähler affine manifold $(Y,g)$ one can define (cf. [KS01]) a characteristic class $[g]$ of the metric, which is an analog of the Kähler class in complex geometry. Let $\mathcal{A}f\negmedspace f_{Y}$ be the sheaf of locally affine functions. The metric is given by local potentials in affine coordinates: $g_{ij}=\frac{\partial^{2}K}{\partial y_{i}\partial y_{j}}$. Then the differences of the potentials on the overlaps will define a Čech cohomology class $[g]\in H^{1}(Y,\mathcal{A}f\negmedspace f_{Y})$.

It is more natural to combine the monodromy representation and the metric class into one class, which we will call the class of affine polarization. It can be represented by a Čech 1-cocycle with values in the semi-direct product $(\operatorname{SL}(n,\mathbb{Z})\ltimes\mathbb{R}^{n})\ltimes\operatorname{Af{}f}_{n}$, where the affine transformations $\operatorname{SL}(n,\mathbb{Z})\ltimes\mathbb{R}^{n}$ act on the affine functions $\operatorname{Af{}f}_{n}$ from the right.

The natural projection onto the normal component $\operatorname{SL}(n,\mathbb{Z})\ltimes\mathbb{R}^{n}$ in the above semi-direct product gives the monodromy representation. To recover the metric class, however, one needs to fix a splitting of the natural map $(\operatorname{SL}(n,\mathbb{Z})\ltimes\mathbb{R}^{n})\ltimes\operatorname{Af{}f}_{n}\to\operatorname{Af{}f}_{n}$. Different splittings will give conjugate metric classes.

For the purposes of this paper we consider a convenient $(n+2)$-dimensional faithful representation of the group $(\operatorname{SL}(n,\mathbb{Z})\ltimes\mathbb{R}^{n})\ltimes\operatorname{Af{}f}_{n}$. Let us choose $q$ – an integral vector in $\mathbb{R}^{n+2}$, and $p$ – an integral vector in the dual space $(\mathbb{R}^{n+2})^{*}$. Then this representation provides an isomorphism of $(\operatorname{SL}(n,\mathbb{Z})\ltimes\mathbb{R}^{n})\ltimes\operatorname{Af{}f}_{n}$ with the following subgroup of $\operatorname{GL}_{n+2}(\mathbb{R})$:

where $g^{*}:(\mathbb{R}^{n+2})^{*}\to(\mathbb{R}^{n+2})^{*}$ is the adjoint linear transformation. The affine space $\mathbb{A}^{n}$ can be identified with $\{\langle p,x\rangle=1\}/q$, and the $G_{n}(p,q)$ action on it gives the corresponding affine transformation of $\mathbb{A}^{n}$. To recover the affine function $f:\mathbb{A}^{n}\to\mathbb{R}$ one needs to fix an integral linear functional $l\in(\mathbb{Z}^{n+2})^{*}$, such that $l(q)=1$. Then $f(x)=l(g(x))-l(x)$ is a function, well defined on the quotient $\{\langle v,x\rangle=1\}/w$. Only the representing Čech cocycle depends on the choice of $l$, not the metric class itself.

For the (mirror) symmetry sake we also choose an integral element $k\in\mathbb{R}^{n+2}$ with $p(k)=1$. The vector $k$ defines an origin in $\mathbb{A}^{n}$, hence it allows to recover the translational part of the affine transformation. Again, the class of this translational part, called the radiance obstruction (cf. [GH84]), is independent of the choice of $k$ (different $k$’s give rise to conjugate monodromies). As was noted in [GS02] the radiance obstruction class is dual to the linear part of the metric class under the duality between the Kähler affine structures.

More generally, it is also clear that the full polarization class for the dual Kähler affine structure can be represented by the adjoint inverse transformations for each $U_{\alpha}\cap U_{\beta}$, with the rôles of $q,k$ and $p,l$ exchanged. Given a basis $\{e_{i}\}$ of $\mathbb{R}^{n+2}$ such that $\langle p,e_{i}\rangle=0,\ i=1,\dots,n+1$, $e_{n+1}=q$ and $e_{n+2}=k$, the group $G_{n}(p,q)$ can be represented by non-degenerate matrices in the form

where $A$ and $b$ represent the linear and translational parts of the affine transformations, and $a,c$ are the linear and constant parts of the affine function, respectively.

All of the above (including the affine structure itself) can be defined even if we do not require the affine charts to be maps into the same affine space. We won’t have groups anymore, but in all cocycle conditions the compositions still make sense. In particular, to specify an integral affine structure we would need continuous maps $\phi_{\alpha}:U_{\alpha}\to\mathbb{A}^{n}_{\alpha}\cong\{\langle p_{\alpha},x_{\alpha}\rangle=1\}/q_{\alpha}$ together with integral elements $q_{\alpha},k_{\alpha}\in\mathbb{R}^{n+2}_{\alpha}$ and $p_{\alpha},l_{\alpha}\in(\mathbb{R}^{n+2})^{*}$ with $\ p_{\alpha}(k_{\alpha})=1,\ l_{\alpha}(q_{\alpha})=1$, such that the transition maps $\phi_{\alpha\beta}:{\mathbb{R}^{n+2}}\to{\mathbb{R}^{n+2}}$ satisfy the corresponding invariance, coinvariance and integrality conditions.

The cohomological information, such as monodromy, radiance obstruction and the metric class, is encoded in the transition maps $\phi_{\alpha\beta}$.

We will be interested in a very special types of integral Kähler affine structures. These structures arise in the metric limits of Calabi-Yau hypersurfaces and complete intersections in toric varieties.

The bi-polyhedral property imposes very severe restrictions on the compatibility between Riemannian metric and affine structure. In particular, $\operatorname{Hess}K_{\alpha}\in\mathcal{L}^{1}_{loc}(\mathbb{R}^{n})$ and the Cauchy-Schwartz inequality implies that the metric completion of $Y$ can be identified with $P$ or $\hat{P}$. This endows both polyhedral complexes with (isomorphic) structures of complete metric spaces.

Next we want to show the existence of bi-PIKAS on ${\Sigma\backslash D}$. Recall from [HZ02] that ${\Sigma\backslash D}$ has a bipartite covering by open sets $U_{v}$ and $V_{w}$. Also, given vectors $\lambda,\nu$ in the interiors of the respective secondary cones $\operatorname{SC}(S),\operatorname{SC}(T)$ with $\lambda(0)=\nu(0)=0$ we can define the polytopes

In the future we will abbreviate the type of a bi-polyhedral integral Kähler affine structure on ${\Sigma\backslash D}$ by simply $(\lambda,\nu)$ having fixed the covering $(\{U_{v},V_{w}\})$.

In order to specify a bi-PIKAS of type $(\lambda,\nu)$ on ${\Sigma\backslash D}$ we will provide the following data. A Legendre dual pair of convex functions $\Phi,\hat{\Phi}$ on $\Delta^{\vee}_{\lambda},{\Delta_{\nu}}$, respectively, smooth on each strata of the respective polytope. (This implies that the Hessians of both $\Phi,\hat{\Phi}$ are positive along the strata.) Then the restrictions of $\Phi$ to the facets of $\Delta^{\vee}_{\lambda}$ serve as potentials for the metric along $U_{v}$. For future use we will prove that it is possible to choose these functions consistently in $\lambda$ and $\nu$.

Note here that adding global linear functions to the potentials $\Phi,\hat{\Phi}$ will induce translations of the polytopes ${\Delta_{\nu}},\Delta^{\vee}_{\lambda}$. Though giving different bi-PIKAS (as we defined them) this will have no effect on the underlying Kähler affine structures (the latter will be canonically equivalent).

Another important observation is that rescaling the data for bi-PIKAS will provide the same metric on ${\Sigma\backslash D}$, though different affine structures.

The metric completion of ${\Sigma\backslash D}$ can be identified with $\Sigma$ and endowed with the structure of a compact metric space via the bi-polyhedral homeomorphisms $\phi,\hat{\phi}$:

Throughout the paper we will often identify points in $\Sigma$, $\partial\Delta^{\vee}_{\lambda}$ and $\partial{\Delta_{\nu}}$ by means of these homeomorphisms when there is no confusion.

We can realize the affine coordinates $y$ on $U_{v}$ and $V_{w}$ explicitly as taking values in the following (affine) subspaces and quotients of $\mathbb{R}^{d}$:

and the transition maps are given by the obvious projections $\mathbb{R}^{d}_{v}(\lambda(v))\to\mathbb{R}^{d}/w$. Then the affine monodromy along a primary loop $(v_{0}w_{0}v_{1}w_{1})$ is given by (cf. [HZ02, Lemma 2.4]):

To describe the full polarization class of a bi-PIKAS of type $(\lambda,\nu)$ on ${\Sigma\backslash D}$ we consider the representation of $(\operatorname{SL}(d-1,\mathbb{Z})\ltimes\mathbb{R}^{d-1})\ltimes\operatorname{Af{}f}_{d-1}$ in $\mathbb{R}^{d}\oplus\mathbb{R}$. The dual space is identified with $(\mathbb{R}^{d})^{*}\oplus\mathbb{R}$. For the charts $U_{v}$ and $V_{w}$ we set

where we have chosen integral elements $l_{w}\in(\mathbb{Z}^{d})^{*}$ and $k_{v}\in\mathbb{Z}^{d}$ such that $\langle l_{w},w\rangle=1$ and $\langle v,k_{v}\rangle=1$. Then for $U_{v}\cap V_{w}$ the cocycle transformation $g_{vw}:\mathbb{R}^{d}\oplus\mathbb{R}\to\mathbb{R}^{d}\oplus\mathbb{R}$ is given by:

where $n^{\prime}=n-\langle v,n\rangle\left[1+\lambda(v)\right]k_{v}$. The cocycle $\{g_{vw}\}$ represents the polarization class which we will denote by $[\lambda,\nu]$.

Then the monodromy representation $\pi_{1}({\Sigma\backslash D})\to\operatorname{GL}(\mathbb{R}^{d}\oplus\mathbb{R})$ along a primary loop $(v_{0}w_{0}v_{1}w_{1})$ is given by

where $\alpha(n)=v_{1}(n)+\lambda(v_{1})v_{0}(n)-[1+\lambda(v)]v_{1}(k_{v_{0}})v_{0}(n)$. Considering this transformation on the quotient by the last coordinate and using $k_{v_{0}}$ to identify $\mathbb{A}^{d-1}=\{\langle v,n\rangle=1\}$ with $\mathbb{R}^{d}_{v}(\lambda(v))=\{\langle v,n\rangle+\lambda(v)=0\}$ via

we recover the above affine monodromy on $\mathbb{R}^{d}_{v}(\lambda(v))$.

Following through the above calculation shows that the converse is also true: any bi-polyhedral Kähler affine structure on ${\Sigma\backslash D}$ in the class $[\lambda,\nu]$ is, in fact, of type $(\lambda,\nu)$.

Among all bi-polyhedral Kähler affine structures of type $(\lambda,\nu)$ we expect to find a unique distinguished representative – the Monge-Ampère structure: in affine coordinates the metric satisfies $\det g_{ij}=c$. Its metric completion to $\Sigma$ is supposed to be the limit of the Ricci-flat metrics on the families of Calabi-Yau hypersurfaces.

Note that the Monge-Ampère constant $c$ is determined from calculating the metric volume of $\Sigma$ as $\sqrt{c}\cdot\operatorname{Vol}\partial\Delta^{\vee}_{\lambda}=\sqrt{c^{-1}}\cdot\operatorname{Vol}\partial{\Delta_{\nu}}$, where $\operatorname{Vol}$ means the affine volume of the corresponding polytopal complex. Also note that the rescaled data $(\epsilon^{-1}\lambda,\epsilon\nu,K_{\alpha}(\epsilon y_{\alpha}),\hat{K}_{\alpha}(\epsilon^{-1}\hat{y}_{\alpha}),\epsilon^{2}g_{ij})$ provides the Monge-Ampère structure in the class $[\epsilon^{-1}\lambda,\epsilon\nu]$ with the same metric on $\Sigma$. Thus the Monge-Ampère bi-PIKAS fit together into a projective family.

We will not address this conjecture any further here, rather we will be happy to start with any bi-polyhedral Kähler affine structure on ${\Sigma\backslash D}$ given by a pair $(\Phi,\hat{\Phi})$.

The $(d-1)$-torus fibration $W(\lambda,\theta)$ (more naturally, a $\mathbb{T}^{d-1}$-torsor) will depend on additional phase multi-parameter $\theta:=\{\theta_{v}\},v\in\operatorname{vert}(S)$, where all $\theta_{v}$ have values in $\mathbb{R}/\mathbb{Z}$. First, we form (trivial) affine torus bundles over the affine open sets by identifying the fibers with the affine tori

The gluing maps are independent of a base point in an overlap $U_{v}\cap V_{w}$ and given there by the natural projection $\mathbb{T}_{v}(\theta_{v})\to\mathbb{T}/w$. This defines the torsor $\pi:W(\lambda,\theta)\to{\Sigma\backslash D}$.

The topology of the total space of the torsor is determined by the combinatorics of $\Sigma$, i.e., independent of $\lambda$ and $\theta$ as long as $\lambda$ is in the right secondary cone $\operatorname{SC}(S)$. In particular, all $W(\lambda,\theta)$ are diffeomorphic to each other, though not canonically.

Since the linear parts of the transition maps are the same for the base and for the fibers, the tangent space at any point in $W(\lambda,\theta)$ splits canonically as

This allows to define a canonical (integrable) almost complex structure on $W(\lambda,\theta)$ as

Given a Riemannian metric $g_{ij}$ on $Y$, one can define the pullback metric $\pi^{*}(g_{ij})$ on $W(\lambda,\theta)$ which is, in fact, Kähler. If, in addition, $g_{ij}$ satisfy the real Monge-Ampère equation, the induced metric on $W(\lambda,\theta)$ is Ricci-flat.

There is another slightly different description of the torus bundle over a Kähler affine manifold $Y$ (cf. [KS01]), which is useful when considering limiting behavior of Calabi-Yau degenerations. We define a torus fibration $W_{\epsilon}(\lambda)$ as the quotient of the total space of the tangent bundle $T({{\Sigma\backslash D}})$ by the integral lattice spanned by $\left\{\epsilon\frac{\partial}{\partial y_{i}}\right\}$, where $y_{i}$ are the affine coordinates. This torus bundle carries canonical complex structure and the pullback metric which comes from the splitting $T_{W_{\epsilon}(\lambda)}\cong T_{\Sigma\backslash D}\oplus T_{\Sigma\backslash D}$ as before.

But in order to make a connection with the previous picture and with the geometry of toric hypersurfaces we need to twist the complex structure on $W_{\epsilon}(\lambda)$ by the element of $H^{1}({\Sigma\backslash D},\mathbb{T}^{d-1})$ associated with the phase parameters $\theta_{v}$. The resulting Kähler manifold $W_{\epsilon}(\lambda,\theta)$ can be canonically identified with $W(\epsilon^{-1}\lambda,\theta)$ constructed by the first method starting with the $\epsilon$-rescaled Kähler affine structure.