3.1. Exponential decay lemma [05CM]

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3.1. Exponential decay lemma

We would like to analyze the behavior of Gibbons-Hawking solutions when the tori (both in the fibers and in the base) are shrinking. First, let us introduce a non-linear differential equation of the Monge-Ampère type.

Definition.

We refer to a pair of real positive definite matrix functions $V^{ij},W^{pq}$ as a solution to the split Monge-Ampère equation in an open subset $R\subset\mathbb{R}^{n}\times\mathbb{R}^{l}$ if $\det V^{ij}=\det W^{pq}$ and they are locally given by a smooth potential function $K$ :

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V^{ij}=\frac{\partial^{2}K}{\partial s_{i}\partial s_{j}},\quad W^{pq}=-\frac{\partial^{2}K}{\partial t_{p}\partial t_{q}},\quad 1\leq i,j\leq n,\quad k+1\leq p,q\leq n+l.

To describe the asymptotics at the discriminant we would like to treat the simplex $\sigma\subset N^{*}$ on the same footing as $\tau$ . Namely, we let $\Sigma\subset N^{*}_{\mathbb{R}}$ be the cone over $\sigma$ , and let $\Sigma^{\vee}$ be its dual cone in $N_{\mathbb{R}}$ . The polyhedral complex $\Pi(\sigma)$ provides a polyhedral decomposition of $N_{\mathbb{R}}/\tau$ into cells $Q_{i}^{\sigma}$ . Denote by $\gamma_{\sigma}$ the $N^{*}_{\tau}$ -valued 1-current defined in the same way as $\gamma_{\tau}$ .

Definition.

Given a domain $R$ in $N^{*}_{\mathbb{R}}/\sigma\times N_{\mathbb{R}}/\tau\cong\mathbb{R}^{n}\times\mathbb{R}^{l}$ a $(\sigma,\tau)$ -type singular solution to the split Monge-Ampère equation in $R$ is a pair of matrix functions $V^{ij},W^{pq}$ which are local Monge-Ampère solutions in $R\setminus(\Pi(\tau)\times\Pi(\sigma))$ with asymptotics at the discriminant locus governed by the distributional equation

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\frac{1}{2\pi}\left(\frac{\partial^{2}W^{pq}}{\partial s_{i}\partial s_{j}}+\frac{\partial^{2}V^{ij}}{\partial t_{p}\partial t_{q}}\right)ds_{i}\wedge dt_{p}=\gamma^{j}_{\tau}(s)\gamma^{q}_{\sigma}(t).

Conjecture 3.1 (Exponential decay lemma).

Given a convex domain $R$ in $\mathbb{R}^{k}\times\mathbb{R}^{l}$ and a $(\sigma,\tau)$ -type solution $V,W$ of the split Monge-Ampère equation in $R$ there is a real one-parameter family of $(\sigma,\tau)$ -solutions $V_{\lambda},W_{\lambda}$ to the Gibbons-Hawking ansatz in $\lambda R\times(S^{1})^{l}$ such that

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The diameter of the circles both in the fiber $T^{n}$ and in the torus part $(S^{1})^{l}$ of the base away from the discriminant is roughly given by $\lambda^{-1}$ .
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The zero Fourier modes of the GH solutions $V_{\lambda}^{0}(u,x),W_{\lambda}^{0}(u,x)$ as functions of the rescaled variables $s,t$ , where $u=\lambda s,x=\lambda t$ , will converge (in some properly weighted norm on the function space) to $V(s,t),W(s,t)$ as $\lambda\to\infty$ .
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The higher Fourier modes decay exponentially away from the discriminant $\Pi(\tau)\times\Pi(\sigma)$ in $\lambda R$ , uniformly in $\lambda$ . That is, if $\beta(u,x)$ denotes the Euclidean distance from the point $(u,x)$ to the discriminant, then

$|V_{\lambda}^{m}(u,x)|\leq C_{1}e^{-\beta(u,x)|m|},\quad|W_{\lambda}^{m}(u,x)|\leq C_{2}e^{-\beta(u,x)|m|},$

for some constants $C_{1},C_{2}$ , and large enough $\lambda$ and $\beta$ .

We would like to give some easy examples and a rough argument based on those why we believe this conjecture is true. Note, however, that once justified, it will have an important consequence for the metric collapse program for the toric hypersurfaces and complete intersections:

Corollary 3.2.

The metric space $(Z_{\sigma,\tau}(\lambda^{-1}R),\lambda^{-2}g_{\lambda})$ , where $g_{\lambda}$ is the Riemannian (orbifold) metric from the Gibbons-Hawking ansatz, converges in the Gromov-Hausdorff sense to $(R,g^{ij}_{\infty})$ , with the limiting metric $g^{ij}_{\infty}=V^{ij}ds_{i}ds_{j}+W^{pq}dt_{p}dt_{q}$ .