Suppose $T^{n}$, a $n$-dimensional (real) torus, acts freely on an $N$-(complex) dimensional Kähler manifold $M$ by Hamiltonian holomorphic isometries. Then $M$ can be considered as a principal $T^{n}$-bundle over a real manifold of dimension $2N-n$. The generalized Gibbons-Hawking ansatz expresses the Kähler and Ricci-flat conditions as differential equations in the $n$ moment map coordinates and $N-n$ holomorphic coordinates on the Kähler quotient.

Let $\mathfrak{t}$ denote the Lie algebra of the torus Lie group $T^{n}$, and let $\mathfrak{t}_{\mathbb{Z}}$ be the natural integral lattice in $\mathfrak{t}$. We will fix a basis in $\mathfrak{t}_{\mathbb{Z}}$. This defines affine coordinates $u_{i}$ on the dual space $\mathfrak{t}^{*}\cong\mathbb{R}^{n}$. Let $Y$ be either $\mathbb{C}^{N-n}$ or $(\mathbb{C}^{*})^{N-n}\cong\mathbb{R}^{N-n}\times(S^{1})^{N-n}$, with the affine complex coordinates $\eta_{p}=x_{p}+iy_{p}$, where $y_{p}$ are the phase coordinates on the torus $(S^{1})^{N-n}$ in the latter case.

Consider a principal $T^{n}$-bundle $\pi:M\to B^{\circ}$ over an open set $B^{\circ}$ in $\mathfrak{t}^{*}\times Y$, with coordinates $(u_{i},\eta_{p},\bar{\eta}_{q})$. Denote by $[\nu]$ its integral Chern class as an element in $H^{2}(B,\mathfrak{t}_{\mathbb{Z}})$.

We are interested in applying the Gibbons-Hawking ansatz to description of the metrics on the toric Calabi-Yau hypersurfaces. Given a torus fibration of such hypersurface near the large complex structure point one may approximate the true Calabi-Yau metric by non-compact solutions of GH equations over different regions of the base. Away from the discriminant locus a semi-flat metric gives a good approximation. We try to argue that as the tori shrink to zero size the metric behavior near singular fibers can also be approximated by certain $y$-periodic solutions of GH ansatz. This local description is the main subject of the paper.

This is an important toy example which provides the local description of Gibbons-Hawking solutions for more interesting cases. Here for the standard toric orbifold metric one can actually write down an explicit solution to the Gibbons-Hawking equations.

First we set up the notations. Let $N\cong\mathbb{Z}^{n+1}$ be an integral lattice in a real vector space $N_{\mathbb{R}}=N\otimes\mathbb{R}$. Denote by $N^{*}\subset N_{\mathbb{R}}^{*}$ the dual lattice in the dual space.

Let $\tau$ be an $n$-simplex with vertices $(w_{0},w_{1},\dots,w_{n})$ in the lattice $N\cong\mathbb{Z}^{n+1}$, whose affine distance from the origin is 1. That is, there is a vector $\rho$ in the dual lattice $N^{*}$ such that $\langle w_{i},\rho\rangle=1$, all $i=0,\dots,n$. Denote by $\mathcal{T}\subset N_{\mathbb{R}}$ the cone over $\tau$ and by $\mathcal{T}^{\vee}\subset N_{\mathbb{R}}^{*}$ the dual cone.

Let $X_{\mathcal{T}}:=\operatorname{Spec}[z^{m}\ :\ m\in\mathcal{T}^{\vee}\cap N^{*}]$ be the associated affine toric variety (cf., e.g. [Ful93]). If $\mathbb{Z}\langle w_{0},\dots,w_{n}\rangle$ denotes the (finite index) sublattice in $N$ generated by $w_{i}$ and $G$ is the quotient group $N/\mathbb{Z}\langle w_{0},\dots,w_{n}\rangle$, then $X_{\mathcal{T}}$ is isomorphic to the orbifold $\mathbb{C}^{k+1}/G$.

The real torus $N_{\mathbb{R}}/N$ acts on $X_{\mathcal{T}}$. But we will be interested rather in the action of its subtorus $T^{n}:=({N_{\rho}})_{\mathbb{R}}/{N_{\rho}}$, where $N_{\rho}:=\{v\in N\ :\ \langle v,\rho\rangle=0\}$. The $n$-dimensional subspace $(N_{\rho})_{\mathbb{R}}\subset N_{\mathbb{R}}$ can be naturally identified with the Lie algebra $\mathfrak{t}$ of $T^{n}$. The dual quotient space $N_{\mathbb{R}}^{*}/\rho$ is identified with ${\mathfrak{t}}^{*}$.

Let $Q^{\tau}_{i}\subset N_{\mathbb{R}}^{*}/\rho$ denote the open normal cones to the vertices $w_{i}$ of $\tau$. Define a polyhedral complex $\Pi(\tau)$ in $N_{\mathbb{R}}^{*}/\rho$ to be the union of walls separating the $Q^{\tau}_{i}$’s:

with the orientation of each wall determined by the ordering of $\{i,j\}$. Another way to look at $\Pi(\tau)$ is as being the image of the union of $(n-1)$-dimensional cones in $\mathcal{T}^{\vee}$ under the quotient map $N_{\mathbb{R}}^{*}\to N_{\mathbb{R}}^{*}/\rho$.

The vector $\rho$ lies in the interior of $\mathcal{T}^{\vee}$, hence $\eta=z^{\rho}:X_{\mathcal{T}}\to\mathbb{C}$ defines a regular function, which vanishes at the divisor in $X_{\mathcal{T}}$ corresponding to the boundary of ${\mathcal{T}}^{\vee}$. Together with the moment map $\mu:X_{\mathcal{T}}\to{\mathfrak{t}}^{*}$ we have the torus fibration

whose restriction to $B^{\circ}=N_{\mathbb{R}}^{*}/\rho\times\mathbb{C}\setminus\Pi(\tau)\times\{0\}$ is a principal $T^{n}$-bundle $\pi:M\to B^{\circ}$.

To describe the topology of this bundle note that the homology group $H_{2}(N_{\mathbb{R}}^{*}/\rho\setminus\Pi(\tau),\mathbb{Z})$ can be naturally identified with $\Lambda_{\tau}$, the (finite index) sublattice of $N_{\rho}$ generated by the elements $w_{i}-w_{j}$ for all pairs of $i,j$. Then the Chern class of this bundle

is the element given by the natural inclusion $\iota:\Lambda_{\tau}\hookrightarrow N_{\rho}$.

The final piece of notation before we describe the standard orbifold metric on $X_{\mathcal{T}}$ is the (finite index) sublattice $N^{\prime}\subset N$ generated by $w_{i}$’s. Let $(N^{\prime})^{*}\supset N^{*}$ be its dual lattice. Let $m_{0},\dots,m_{n}$ be the minimal vectors in $(N^{\prime})^{*}$ along the rays of ${\mathcal{T}}^{\vee}$.

In polar coordinates the standard orbifold metric on the algebraic torus $(\mathbb{C}^{*})^{n+1}\subset X_{\mathcal{T}}$ will be

The functions $z^{m_{i}}$ are defined only on the $|G|$-fold covering space of $X_{\mathcal{T}}$, but $|z|^{m_{i}}$ are well defined on $X_{\mathcal{T}}$ itself. So are the differential forms $\langle m_{i},d\theta\rangle$.

To write this metric in the Gibbons-Hawking ansatz we choose a basis $\{e_{i}\}$ of $N_{\rho}=\mathfrak{t}$. Evaluating the moment map on the basis vectors defines the coordinates $u_{i}=\mu(e_{i})$ on $N_{\mathbb{R}}^{*}/\rho={\mathfrak{t}}^{*}$, thus giving an identification $N_{\mathbb{R}}^{*}/\rho$ with $\mathbb{R}^{n}$. The metric on each phase torus $T_{|z|}:=\{z\ :\ |z|^{m_{i}}=const\}\cong N_{\mathbb{R}}/N$ is constant, and, hence, it is given by a quadratic form $Q_{|z|}$ on $N_{\mathbb{R}}$. Let $(V^{-1})^{ij}$ be the matrix of restriction of $Q_{|z|}$ to $(N_{\rho})_{\mathbb{R}}$ in the basis $\{e_{i}\}$. Then the functions $V^{ij}$ and $W=\det V^{ij}$ give a solution to the GH equations.

Note that the top degree holomorphic form $\Omega_{\tau}$ coincide with the push forward under the projection $\mathbb{C}^{n+1}\to\mathbb{C}^{n+1}/G$ of the standard volume form on $\mathbb{C}^{n+1}$.

As an illustration free of orbifold complications let us write the ansatz for the standard Euclidean metric on $\mathbb{C}^{n+1}$ explicitly. In this case, $\tau$ is the standard $n$-simplex in $N:=\mathbb{Z}^{n+1}$, i.e. $w_{i}$ form a basis in $N$. We will fix the coordinates $z_{i}=|z_{i}|e^{i\theta_{i}}$ on $\mathbb{C}^{n+1}$. The action of the torus

on $\mathbb{C}^{n+1}$ gives rise to a principal $T^{n}$-bundle over $M=\mathbb{R}^{n}\times\mathbb{C}\setminus\Pi(\tau)\times\{0\}$. Then, in the Gibbons-Hawking coordinates the metric on $M$ can be written as

where

The above expressions degenerate whenever two or more of the coordinates $z_{i}$ vanish. Thus, the discriminant locus $D\subset\mathbb{R}^{n}\times\mathbb{C}$ is given by $u\in\Pi(\tau)$ and $\eta=0$. However, when written in the Euclidean coordinates the metric extends from $M$ to the standard flat metric $h=\sum_{i=0}^{n}dz_{i}\otimes d\bar{z}_{i}$ on $\mathbb{C}^{n+1}$.

We start with a $T^{n}$-torus bundle $\pi:M\to R^{n}\times\mathbb{C}\setminus\Pi(\tau)\times\{0\}$ which has the same topology as the orbifold bundle above. It is convenient to encode the topological information about the bundle by rewriting the equation (5) for the curvature in a distributional form. Let $\gamma^{j}_{\tau}(u)$ be the $N_{\rho}$-valued 1-current supported on $\Pi(\tau)$ defined by

for an $(n-1)$-form $\alpha$. Then adding the distributional equation

to the Gibbons-Hawking ansatz will automatically guarantee that the fibration $\pi:M\to R^{n}\times\mathbb{C}\setminus\Pi(\tau)\times\{0\}$ has the right Chern class. Here $\delta(\eta)$ stands for the two-current associated to the origin in $\mathbb{C}$ (the Dirac delta-function).

A remark on notation: $\gamma_{\tau}$ and $\delta$ in (10) really mean the pull back of the corresponding currents to the product $\mathbb{R}^{n}\times\mathbb{C}$. We will continue to abuse this notation throughout the rest of the paper when there is no confusion possible.

It is an interesting problem to exhibit the existence (and abundance) of the Gibbons-Hawking solutions. For instance, one can try to deform the standard (flat) orbifold metric. Applying continuity method techniques (cf. [GT83, Ch. 17]) this amounts to inverting a second order linear elliptic differential operator – the linearization of the GH operator at the flat solution. The difficulty is that one of the eigenvalues of its symbol blows off as we approach the discriminant.

In two dimensions (the original GH ansatz [Haw77],[GH78]) the equation becomes the usual Laplace equation. Since there are no positive harmonic functions on $\mathbb{R}^{3}$ except constants, any solution has the form

where $\ell(\tau)$ is the length of $\tau$, and $a$ is a positive constant. This defines the famous Taub-NUT metric – the first example of a non-trivial complete Kähler Ricci-flat metric on $\mathbb{C}^{2}$ and its quotients by cyclic groups.

The goal here is to set up the Gibbons-Hawking ansatz in such a way that the resulting complex manifold is identifiable with the local model for a Calabi-Yau toric hypersurface. There are no explicit solutions known in dimension higher than 2, unlike the orbifold case. But we will try to make use of the ansatz to get some information about the limiting behavior of solutions in certain degenerations.

We will adapt the notations from the orbifold example. Namely, $\tau=\{w_{0},\dots,w_{n}\}$ is a simplex (but now not necessarily of codimension 1) in the lattice $N\cong\mathbb{Z}^{n+l+1}$, which has the distance 1 from the origin. Let $\sigma=\{v_{0},\dots,v_{l}\}$ be a simplex in $N^{*}$ such that $\langle\sigma,\tau\rangle=1$. In particular, it means that $\sigma$ also has distance 1 from the origin. Let $N_{\sigma}\subset N$ and $N^{*}_{\tau}\subset N^{*}$ be the sublattices orthogonal to $\sigma$ and $\tau$, respectively. And let

be the corresponding (dual) quotients. The polyhedral complex $\Pi(\tau)$ provides a polyhedral decomposition of $N^{*}_{\mathbb{R}}/\sigma$ into cells $Q_{i}^{\tau}$ and $\gamma_{\tau}$ is the associated 1-current supported on $\Pi(\tau)$, as before. Also, we define the cone $\mathcal{T}:=\operatorname{cone}(\tau)$ in $N_{\mathbb{R}}$, its dual $\mathcal{T}^{\vee}\subset N^{*}_{\mathbb{R}}$ and let

be the associated affine toric variety.

Every vertex $v_{i}\in\sigma$ lies in the interior of $\mathcal{T}^{\vee}$, and, hence the monomials $z^{v_{i}}$ belong to the coordinate ring of $X_{\mathcal{T}}$. Let $Z_{\sigma,\tau}$ denote the closure of the affine hypersurface

in the toric variety $X_{\mathcal{T}}$.

The real torus $T^{n}:=(N_{\sigma})_{\mathbb{R}}/N_{\sigma}$ acts on $X_{\mathcal{T}}$ and leaves the hypersurface $Z_{\sigma,\tau}$ invariant. We assume that this action is a holomorphic isometry and denote by $\mu:Z_{\sigma,\tau}\to N_{\mathbb{R}}^{*}/\sigma$ the corresponding moment map.

The natural inclusion $N^{*}_{\tau}\subset\mathcal{T}^{\vee}\cap N^{*}$ gives the projection $\kappa:X_{\mathcal{T}}\to T_{\tau}$ onto the algebraic torus $T_{\tau}:=\operatorname{Spec}[z^{m}\ :\ m\in N^{*}_{\tau}]$. A choice of $\rho\in N^{*}$, such that $\langle\rho,\tau\rangle=1$, defines the polynomial

which can be thought of as a polynomial in $T_{\tau}$. The zero divisor of $P_{\sigma}$ does not depend on the choice of $\rho$ and let $\Gamma_{\sigma}$ denote the 2-current in $T_{\tau}$ associated to it.

We will consider the map $(\mu,\kappa):Z_{\sigma,\tau}\to N_{\mathbb{R}}^{*}/\sigma\times T_{\tau}$ as a torus fibration with the discriminant locus $D=\Pi(\tau)\times\{P_{\sigma}(z)=0\}$. When restricted to a domain $B\subset N_{\mathbb{R}}^{*}/\sigma\times T_{\tau}$ it defines a torus fibration $Z_{\sigma,\tau}(B)\to B$ which is a principal $T^{n}$-bundle over $B^{\circ}:=B\setminus D$. The Chern class is given, as before, by the inclusion $\iota:\Lambda_{\tau}\hookrightarrow N_{\sigma}$.

If the torus action is a holomorphic isometry a Ricci-flat metric on $Z_{\sigma,\tau}(B)$ can be written in the Gibbons-Hawking form. Our main goal of this section is to prove the converse. That is if we have a GH solution with the right Chern class then it defines a Ricci-flat metric on $Z_{\sigma,\tau}(B)$.

To write everything in coordinates we choose a basis $\{e_{i}\}$ in $N_{\sigma}$ and a basis $\{m_{p}\}$ in $N^{*}_{\tau}$. This will defines the coordinates $u_{i}:=\mu(e_{i})$ on $N_{\mathbb{R}}^{*}/\sigma\cong\mathbb{R}^{n}$ and $\eta_{p}:=\log(z^{m_{p}})=\langle m_{p},\log z\rangle$ on $T_{\tau}\cong(\mathbb{C}^{*})^{l}$.

The topological information about the bundle is again encoded in right hand side of the equation (16).

To state the compatibility with the desired holomorphic volume form we recall (cf., e.g., [Bat93]) that given an affine hypersurface $Z_{f}=\{f(z)=0\}\subset(\mathbb{C}^{*})^{N+1}$ there is a distinguished top degree holomorphic form $\Omega_{CY}$ on $Z_{f}$, which is defined as a Poincaré residue of the meromorphic $(N+1)$-form

on $(\mathbb{C}^{*})^{N+1}$ with a single pole along $Z_{f}$. This form is special in the following sense. If the hypersurface $Z_{f}$ is compactified to a Calabi-Yau hypersurface in a projective toric variety, then $\Omega_{CY}$ is the restriction of the unique (up to a scalar multiple) non-vanishing holomorphic volume form on the Calabi-Yau manifold (orbifold).

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.