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

$$N_{\mathbb{R}}^{*}/\sigma:=N_{\mathbb{R}}^{*}/\langle v_{0},\dots,v_{l}\rangle,\quad N_{\mathbb{R}}/\tau:=N_{\mathbb{R}}/\langle w_{0},\dots,w_{n}\rangle$$

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

$$X_{\mathcal{T}}:=\operatorname{Spec}[z^{m}\ :\ m\in\mathcal{T}^{\vee}\cap N^{*}]$$

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

$$\{z\in(\mathbb{C}^{*})^{k+l+1}\ :\ \sum_{i=0}^{l}z^{v_{i}}=1\}$$

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

$$P_{\sigma}(z):=z^{-\rho}\sum_{i=0}^{l}z^{v_{i}},$$

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

$$\frac{dz_{0}\wedge\dots\wedge dz_{N}}{f\cdot z_{0}\dots z_{N}}$$

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).