2.2. Bi-polyhedral Kähler affine structures [03EQ]

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2.2. Bi-polyhedral Kähler affine structures

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.

Definition.

An integral affine structure on $Y$ is polyhedral if there is an $n$ -dimensional polyhedral complex $P$ , a collection of disjoint open sets $\{U_{\alpha}\}$ , whose closures cover $Y$ , i.e. $Y=\bigcup\overline{U}_{\alpha}$ , and a continuous map $\phi:Y\to P$ , which provides an affine homeomorphism of each $U_{\alpha}$ with the interior of some $n$ -dimensional face of $P$ . We say that the pair $(\{U_{\alpha}\},P)$ realizes the polyhedral affine structure if $P$ is minimal, which, in particular, means that there is a bijection between open sets $\{U_{\alpha}\}$ and $n$ -dimensional cells of $P$ .

Definition.

An integral Kähler affine structure on $Y$ is bi-polyhedral (bi-PIKAS for short) if there is a bipartite covering $\{U_{\alpha},V_{\beta}\}$ of $Y$ and two polyhedral complexes $P,\hat{P}$ such that $(\{U_{\alpha}\},P)$ and $(\{V_{\beta}\},\hat{P})$ provide polyhedral realizations of the underlying affine structure and its dual, respectively. We say that the bi-polyhedral Kähler affine structure is of type $(\{U_{\alpha},V_{\beta}\},P,\hat{P})$ .

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

	$\displaystyle\Delta^{\vee}_{\lambda}=\{n\in\mathbb{R}^{d}\ :\ \langle m,n\rangle+\lambda(m)\leq 0\text{ for all }m\in\Delta_{\mathbb{Z}}\},$
	$\displaystyle{\Delta_{\nu}}=\{m\in(\mathbb{R}^{d})^{*}\ :\ \langle m,n\rangle+\nu(n)\leq 0\text{ for all }n\in\Delta^{\vee}_{\mathbb{Z}}\}.$

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 $\nu$ .

Definition.

Suppose, for each pair $(\lambda,\nu)\in\operatorname{SC}(S)\times\operatorname{SC}(T)$ we have a bi-polyhedral Kähler affine structure on ${\Sigma\backslash D}$ of type $(\lambda,\nu)$ , which varies continuously with $(\lambda,\nu)$ in the Hausdorff topology of metric structures on $\Sigma$ . We call such a family projective if:

•

For any linear functions $l\in\mathbb{R}^{d},\ l^{\vee}\in(\mathbb{R}^{d})^{*}$ , the bi-PIKAS for $(\lambda+l,\nu+l^{\vee})\in\operatorname{SC}(S)\times\operatorname{SC}(T)$ have the same underlying Kähler affine structure.

•

The bi-PIKAS for $(\epsilon^{-1}\lambda,\epsilon\nu)$ differs from the bi-PIKAS for $(\lambda,\nu)$ by the $\epsilon$ -rescaling

(y^{\prime}_{\alpha},\hat{y}^{\prime}_{\alpha},K^{\prime}_{\alpha}(y^{\prime}_{\alpha}),\hat{K}^{\prime}_{\alpha}(\hat{y}^{\prime}_{\alpha}),g_{ij})=(\epsilon^{-1}y_{\alpha},\epsilon\hat{y}_{\alpha},K_{\alpha}(\epsilon y^{\prime}_{\alpha}),\hat{K}_{\alpha}(\epsilon^{-1}\hat{y}^{\prime}_{\alpha}),\epsilon^{2}g_{ij})

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.

Proposition 2.1.

There are Legendre dual functions $\Phi\colon\Delta^{\vee}_{\lambda}\rightarrow\mathbb{R}$ and $\hat{\Phi}\colon{\Delta_{\nu}}\rightarrow\mathbb{R}$ that define a projective family of bi-PIKAS on ${\Sigma\backslash D}$ .

Proof.

First, we choose a smooth function with positive Hessian on $\Delta_{\lambda-\beta}^{\vee}$ whose gradients stay in $\Delta_{\nu-\beta}$ , and cover $\Delta_{\nu-2\beta}$ .

$[Uncaptioned image]$ $[Uncaptioned image]$

Figure 1: The domain for the first step. The set of gradients.

As a second step, we need a continuous strictly convex function $\tilde{\nu}$ on $\beta\Delta^{\vee}$ that is an approximation of the function $\nu$ with the following properties.

•

$\tilde{\nu}$ is piecewise smooth.
•

$\operatorname{Hess}\tilde{\nu}>0$ on the smooth pieces.
•

The gradients along $\operatorname{cone}\tau$ belong to a neighborhood of the corresponding vertex in ${\Delta_{\nu}}$ that are pairwise disjoint, and do not meet the $\beta$ neighborhood of the barycenter of ${\Delta_{\nu}}$ .
•

For a vertex $w\in T$ , the $w$ -directional derivatives equal $-\nu(w)$ in the star neighborhood of $\beta w$ in the barycentric subdivision of $\beta T$ .

$[Uncaptioned image]$

Figure 2: The set of gradients of $\tilde{\nu}$ .

We obtain a convex function on $(\mathbb{R}^{d})^{*}$ if we consider the lower hull of the ( $d+1$ )-dimensional Minkowski sum of graphs of the two functions.

Finally, we want to obtain a function that is smooth along the strata. As explained in § 3.2, we convolute with a kernel that depends on the position $x\in\Delta^{\vee}_{\lambda}$ as follows. Consider the quadratic form

Q(x)=\sum_{v\in S}Q_{v}(x)\ \text{ where }\ Q_{v}(x)=\frac{1}{\rho_{v}(x)}v^{2}

This quadratic form is non-degenerate because the $v$ ’s span $(\mathbb{R}^{d})^{*}$ . It has a dominant summand if $x$ is close to a facet. Now our kernel will be a normalized $e^{1-Q(x)}$ . Its support – the ellipsoid given by $Q(x)\leq 1$ – depends on the position as sketched in the figure.

$[Uncaptioned image]$

Figure 3: The support of the mollifier.

The obtained function will have a positive Hessian along the strata, so that the Legendre dual function will be smooth along its corresponding strata. ∎

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 $\mathbb{R}^{d}$ :

	$\displaystyle y(q)\in\mathbb{R}^{d}_{v}(\lambda(v)):=\{n\in\mathbb{R}^{d}\ :\ \langle v,n\rangle+\lambda(v)=0\},\quad q\in U_{v},$
	$\displaystyle y(q)\in\mathbb{R}^{d}/w,\quad q\in V_{w},$

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]):

(1)

n\mapsto n+[\langle v_{1},n\rangle+\lambda(v_{1})](w_{1}-w_{0}),\quad n\in\mathbb{R}^{d}_{v}(\lambda(v)).

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

	$\displaystyle U_{v}:\ p=\left(\begin{array}[]{c}0\\ 1\end{array}\right),\ k=\left(\begin{array}[]{c}k_{v}\\ 0\end{array}\right),\ q=(v,0),\ l=(0,1),$
	$\displaystyle V_{w}:\ p=\left(\begin{array}[]{c}w\\ 0\end{array}\right),\ k=\left(\begin{array}[]{c}0\\ 1\end{array}\right),\ q=(0,1),\ l=(l_{w},0),$

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:

\left(\begin{array}[]{c}n\\ s\end{array}\right)\mapsto\left(\begin{array}[]{c}n^{\prime}-\left[1+\nu(w)\right]\langle l_{w},n^{\prime}\rangle w+sw\\ \langle v,n\rangle\end{array}\right),

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

\left(\begin{array}[]{c}n\\ s\end{array}\right)\mapsto\left(\begin{array}[]{c}n\\ s\end{array}\right)+\alpha(n)\left(\begin{array}[]{c}w_{1}-w_{0}\\ \nu(w_{1})-\nu(w_{0})\end{array}\right),

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

\displaystyle n\mapsto n-\langle v,n\rangle\left[1+\lambda(v)\right]k_{v},

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 $(\lambda,\nu)$ .