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.

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 $\gamma_{\tau}$.

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:

We consider the case when either $l=0$, or $k=0$. In both situations the discriminant locus is empty and the total space $M$ is just the product of the domain $R$ and the torus $T^{n}$. We can use any solution of the classical real Monge-Ampère equation in $R$ and extend it to a Gibbons-Hawking solution on $M$ by setting higher Fourier modes to zero. In the obvious complex structure this will give a Ricci-flat metric on $M=\bar{M}$ (cf. [Hit97], [Leu00], [LYZ01]).

This is the periodic version of the original Gibbons-Hawking ansatz [GH78],[Haw77]. We consider the case when $k=l=1$ and both simplices $\tau$ and $\sigma$ are of length 1, although the construction works for a non-unimodular case as well.

The Gibbons-Hawking equation in this case is equivalent to the Laplace equation for $V(u,x,y)$ ($=W(u,x,y)$) on a domain in the cylinder $\mathbb{R}\times\mathbb{R}\times S^{1}$ with the Dirac $\delta$-function on the right hand side. We can write both the solution $V(u,x,y)$ and the $\delta$-function in the Fourier expansion:

$$V(u,x,y)=\sum_{m\in\mathbb{Z}}V_{m}e^{2\pi imy},\quad\gamma^{j}_{\tau}(u)\delta_{P_{\sigma}}(\eta)=-\delta(u,x,y)=-\sum_{m\in\mathbb{Z}}\delta(u,x)e^{2\pi imy}.$$

Here the minus sign takes into account the orientation of the circle action when passing from currents to generalized functions.

Being linear, the Gibbons-Hawking equation

$$\frac{\partial^{2}V_{\lambda}}{\partial u^{2}}+\frac{\partial^{2}V_{\lambda}}{\partial x^{2}}+\frac{\partial^{2}V_{\lambda}}{\partial y^{2}}=-2\pi\cdot\delta(u,x,y)$$

will decompose into the Helmholtz equations according to the Fourier modes:

$$\frac{\partial^{2}V_{\lambda}^{m}}{\partial u^{2}}+\frac{\partial^{2}V_{\lambda}^{m}}{\partial x^{2}}-(2\pi m)^{2}V_{\lambda}^{m}=-2\pi\cdot\delta(u,x),\quad m\in\mathbb{Z}.$$

On the other hand, the $(\sigma,\tau)$-type split Monge-Ampère equation in the rescaled coordinates $s=\lambda^{-1}u,t=\lambda^{-1}u$ is the two-dimensional Laplace equation:

$$\frac{\partial^{2}V}{\partial s^{2}}+\frac{\partial^{2}V}{\partial t^{2}}=-2\pi\cdot\delta(s,t),$$

whose fundamental solutions are in the form $V(s,t)=-\frac{1}{2}\log|s^{2}+t^{2}|+h(s,t)$, for a harmonic function $h$. Thus, one can take the zero mode of the corresponding Gibbons-Hawking solution to be $V_{\lambda}^{0}(u,x)=V(\lambda^{-1}u,\lambda^{-1}x)$, as long as $V(s,t)$ stays positive on $R$. As for the higher modes, it is known that a fundamental solution to the Helmholtz equation with $m\neq 0$ may be given by the Bessel function

$$V_{\lambda}^{m}=K_{0}(2\pi|m|r)\sim\frac{1}{\sqrt{4|m|r}}\,e^{-{2\pi|m|r}}\left(1+O(r^{-1})\right),\text{ where }r^{2}=u^{2}+x^{2},$$

which decays exponentially as required.

The full proof of the conjecture in this general case will probably require some very non-trivial application of the continuity method to deform the given split solution, then introduce exponentially small higher modes and do some clever estimates afterwards. Meanwhile, we want to indicate a rough argument why some of the ideas from the K3 example above may still work in general.

To have the Fourier modes of the solutions defined on the same domain, independent of $\lambda$, we can scale the variables by $\lambda$:

$$s_{i}=\frac{u_{i}}{\lambda},\quad t_{p}=\frac{x_{p}}{\lambda},\quad\tilde{y_{p}}=\frac{y_{p}}{\lambda}.$$

Then the GH solutions are on $R\times(S^{1}/\lambda)^{l}$ for all $\lambda$, and their Fourier modes are functions on $R$. To keep up with the complex structure the logarithmic map $(\mathbb{C}^{*})^{l}\to\mathbb{R}^{l}$ has to scale by $\lambda$ as well:

$$\log_{e^{\lambda}}(z_{1},\dots,z_{n}):=\frac{1}{\lambda}(\log|z_{1}|,\dots,\log|z_{n}|).$$

We would like to recall a few basic facts from “tropical” geometry (cf., e.g., [Mik01]). Given a polynomial $P_{\sigma}(z)$ in $(\mathbb{C}^{*})^{l}$ the amoeba $\mathcal{A}^{\lambda}_{\sigma}$ is defined to be the image of the rescaled log map:

$$\mathcal{A}^{\lambda}_{\sigma}:={\log_{e^{\lambda}}(\{P_{\sigma}=0)\}}\subset\mathbb{R}^{l}.$$

As $\lambda\to\infty$ the amoeba $\mathcal{A}^{\lambda}_{\sigma}$ approaches its spine $\mathcal{A}^{\infty}_{\sigma}=\Pi(\sigma)$. The Ronkin function

$$N_{\sigma}(x):=\frac{1}{(2\pi\sqrt{-1})^{l}}\int\limits_{\log|z|=x}\!\log|P_{\sigma}(z)|^{2}\ \frac{dz_{1}}{z_{1}}\wedge\dots\wedge\frac{dz_{l}}{z_{l}}$$

is defined up to a linear function, which depends on a particular choice of $\rho\in N^{*}$ used in the definition of the polynomial $P_{\sigma}$. Denote by

$$N_{\sigma}^{\lambda}(t):=\frac{1}{\lambda}N_{\sigma}(\lambda t)$$

the rescaled Ronkin function. The point is that $N^{\lambda}_{\sigma}(t)$ is a continuous function, linear on each connected component of $\mathbb{R}^{l}\setminus A^{\lambda}_{\sigma}$, with the slopes given by the $v_{i}$’s. As $\lambda\to\infty$, it converges to the piece-wise linear function $N_{\sigma}^{\infty}(t)$ whose corner locus is $\Pi(\sigma)$ with the $v_{i}$-slopes over the $Q^{\sigma}_{i}$.

We would like to analyze the right hand side of the equation (16) written in the Fourier expansion. The factor $\gamma_{\tau}$ carries over to every mode, while

$$\Gamma_{\sigma}=\frac{\sqrt{-1}}{2\pi}\partial\bar{\partial}\log|P_{\sigma}|^{2}$$

decomposes into currents supported on $\mathcal{A}^{\lambda}_{\sigma}$. In particular, since the exterior differentiation commutes with averaging, we conclude that the zero mode of $\Gamma_{\sigma}$ is given by the Hessian of the Ronkin function:

$$\Gamma_{\sigma}^{0}=\frac{1}{(2\pi\sqrt{-1})^{l}}\!\int\limits_{\log|z|=x}\!\Gamma_{\sigma}\ \frac{dz_{1}}{z_{1}}\wedge\dots\wedge\frac{dz_{l}}{z_{l}}=\\ \frac{\sqrt{-1}}{2\pi}\cdot\frac{1}{(2\pi\sqrt{-1})^{l}}\ \partial\bar{\partial}\!\int\limits_{\log|z|=x}\!\log|P_{\sigma}|^{2}\ \frac{dz_{1}}{z_{1}}\dots\frac{dz_{l}}{z_{l}}=\frac{\partial^{2}N_{\sigma}(x)}{\partial x_{p}\partial x_{q}}dx_{p}\wedge dy_{q}.$$

But substituting $x=\lambda t$ yields

$$\frac{\partial^{2}N_{\sigma}(x)}{\partial x_{p}\partial x_{q}}dx_{p}=\frac{\partial^{2}N^{\lambda}_{\sigma}(t)}{\partial t_{p}\partial t_{q}}dt_{p}.$$

Hence, as $\lambda\to\infty$, the current $\Gamma_{\sigma}^{0}$ converges to $\gamma_{\sigma}dy_{q}$.

As for the higher modes, we note that since $\tilde{y}$ is now $(2\pi\lambda^{-1})$-periodic, there is a factor of $\lambda^{2}$ in the zero order term of the Helmholtz-type equation for $m\neq 0$. By analogy with the Bessel functions we hope that the spectral theory will force the higher modes decay exponentially away from the locus $\mathcal{A}^{\lambda}_{\sigma}\times\Pi(\tau)$ with the exponent now multiplied by an arbitrary large number $\lambda$.

One can go about proving the lemma by starting with the given split Monge-Ampère solution and constructing a family of solutions but with the factor $\gamma_{\sigma}dy_{q}$ in the right hand side being replaced by a more regular $\Gamma_{\sigma}^{0}$. This will give a family of semi-flat Gibbons-Hawking solutions. Then one can argue that since the higher modes can be taken exponentially small, they may be considered, in some sense, as perturbation of the semi-flat solution.