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:

$$\Pi(\tau):=\bigcup_{i\neq j}\mathrm{wall}_{ij},$$

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

$$(\mu,\eta):X_{\mathcal{T}}\to N_{\mathbb{R}}^{*}/\rho\times\mathbb{C},$$

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

$$[\nu]\in H^{2}(N_{\mathbb{R}}^{*}/\rho\setminus\Pi(\tau),\mathbb{Z})\otimes N_{\rho}\cong\operatorname{Hom}(\Lambda_{\tau},N_{\rho})$$

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

$$h=\sum_{i=0}^{n}(d|z^{m_{i}}|+\sqrt{-1}|z^{m_{i}}|\langle m_{i},d\theta\rangle)\otimes(d|z^{m_{i}}|-\sqrt{-1}|z^{m_{i}}|\langle m_{i},d\theta\rangle).$$

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

$$T^{n}=\{(\theta_{0},\dots,\theta_{n})\ :\ \sum\theta_{i}=0\}$$

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

$$h=(V^{-1})^{ij}(V^{ki}du_{k}+\sqrt{-1}\cdot A_{i})\otimes(V^{kj}du_{k}-\sqrt{-1}\cdot A_{j})+Wd\eta\otimes d\bar{\eta},$$

where

	$$\displaystyle u_{i}=|z_{i}|^{2}-|z_{0}|^{2},\ i=1,\dots,n,\qquad\eta=z_{0}z_{1}\dots z_{n},$$
	$$\displaystyle W^{-1}=|z_{0}z_{1}\dots z_{n}|^{2}\left(\frac{1}{|z_{0}|^{2}}+\frac{1}{|z_{1}|^{2}}+\dots+\frac{1}{|z_{n}|^{2}}\right),$$
	$$\displaystyle(V^{-1})^{ij}=|z_{0}|^{2}+\delta^{ij}|z_{i}|^{2},$$
	$$\displaystyle A_{j}=d\theta_{j}-W|z_{0}z_{1}\dots\widehat{z_{k}}\dots z_{n}|^{2}\cdot d(\theta_{0}+\theta_{1}+\dots+\theta_{n}).$$

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 $\mathbb{C}^{n+1}$.