We begin by recalling the familiar theory in complex dimension two. In this case Calabi-Yau metrics are locally hyperkähler and such metrics with circle symmetry are locally given by the classical Gibbons-Hawking ansatz, in terms of a positive harmonic function on a domain in $\mathbb{R}^{3}$. Notice however in the usual Gibbons-Hawking construction the hyperkähler metrics admits an $S^{2}$ family of parallel compatible complex structures and there is a priori no preferred choice, but if we do make a choice of complex structure the base $\mathbb{R}^{3}$ also has a natural splitting into $\mathbb{C}\oplus\mathbb{R}$, and we refer to Section 2.3 for further discussion. Fixed points of the circle action correspond to simple poles of the harmonic function, i.e. Dirac type singularities, locally given by $\frac{1}{2r}$ plus a smooth function. Local topological model for the $S^{1}$ fibration near a singularity is the standard Hopf fibration $\pi:\mathbb{R}^{4}\rightarrow\mathbb{R}^{3}$. Applying the Gibbons-Hawking construction to the entire $\mathbb{R}^{3}$ with the harmonic function $\frac{1}{2r}+C(C>0)$, one obtains a homothetic scaling family of the Taub-NUT metrics on $\mathbb{R}^{4}$, which limits to the flat $\mathbb{R}^{4}$ when $C\rightarrow 0$, and to the flat $\mathbb{R}^{3}$ when $C\rightarrow\infty$. Now we can also apply the Gibbons-Hawking ansatz to flat three manifolds with slower volume growth, but then one can not expect a non-trivial global positive harmonic function with only simple poles, nevertheless the construction still yields very interesting family of incomplete hyperkähler metrics. Important examples are given by the Green’s function on $S^{1}\times\mathbb{R}^{2}$ (the Ooguri-Vafa metric, c.f. [GW00]) and $T^{2}\times\mathbb{R}$ (c.f. [HSVZ18]). These metrics are important in understanding the collapsing behavior of hyperkähler metrics on K3 surfaces [GW00, HSVZ18].

In general one expects that when collapsing occurs for a family of hyperkähler metrics on K3 surfaces, certain nilpotent fibration structure should appear and due to topological reasons singular fibers often have to appear. The above incomplete metrics are adapted to model the collapsing near the singular fibers, and they exhibit interesting multi-scale collapsing phenomenon.

In higher dimensions, algebro-geometric consideration concerning complex structure degenerations suggests the significance of Calabi-Yau metrics with torus symmetry. Our basic observation is that suppose we have a degenerating family $\mathcal{N}\rightarrow\Delta\subset\mathbb{C}$ of smooth complex algebraic varieties $\mathcal{N}_{t}$ into $\mathcal{N}_{0}$ which is a union of irreducible components. Then in generic situation, near a point on $\mathcal{N}_{0}$ where $k+1$ components intersect transversally, the degeneration family is locally modeled by an equation of the form

(2.1)

$$z_{0}\cdots z_{k}=t(f(z_{k+1},\cdots z_{n})+g)$$

where $g$ is contained in the analytic ideal generated by $z_{0},\cdots,z_{k}$. Near a point with $z_{0}=\cdots=z_{k}=0$, this can be further approximated by omitting the term $g$, which results in a $(\mathbb{C}^{*})^{k}$ fibration

(2.2)

$$z_{0}\cdots z_{k}=tf(z_{k+1},\cdots,z_{n})$$

over a $n-k$ dimensional base. The fibers are orbits of the $(\mathbb{C}^{*})^{k}$ action, where $(\mathbb{C}^{*})^{k}$ is naturally a subgroup in $(\mathbb{C}^{*})^{k+1}=\{(\lambda_{0},\cdots,\lambda_{k})|\lambda_{i}\in\mathbb{C}^{*}\}$ defined by the relation $\lambda_{0}\cdots\lambda_{k}=1$.

Slightly more globally one can consider a complex manifold $D$ and $k+1$ holomorphic line bundles $L_{0},\cdots,L_{k}$ over $D$. Denote the vector bundle $E=\oplus L_{j}$. Fix a holomorphic section $f$ of the tensor product $L_{0}\otimes\cdots\otimes L_{k}\simeq\det(E)$. Then we can consider the hypersurface $\mathcal{N}$ in $E\times\mathbb{C}$ cut-out by the equation

(2.3)

$$s_{0}\otimes\cdots\otimes s_{k}=tf(x)$$

where $(x,[s_{0},\cdots,s_{k}])$ is a point in $E$ and $t\in\mathbb{C}$. We can view $\mathcal{N}$ as a family of hypersurfaces in $E$ parametrized by $t\in\mathbb{C}$. There is a natural $\mathbb{C}^{*}$ action on $\mathcal{N}$ given by

(2.4)

$$\lambda(\zeta).(x,[s_{0},\cdots,s_{k}],t)=[x,[\zeta s_{0},\cdots,\zeta s_{k}],\zeta^{k+1}t)$$

It induces isomorphisms between $\mathcal{N}_{t}$ and $\mathcal{N}_{1}$ for all $t\neq 0$, and it preserves $\mathcal{N}_{0}$.

For simplicity we only consider the generic case when the zeroes of $f$ form smooth hypersurface, then for $t\neq 0$, $\mathcal{N}_{t}$ is smooth but the projection map $\pi_{t}:\mathcal{N}_{t}\rightarrow D$ is still singular precisely along the union of $\Pi_{ij}\equiv\{x\in D|s_{i}(x)=s_{j}(x)=0\}$ for all pairs $(i,j)$ with $i\neq j$. Notice this union is also the singular set of the total space $\mathcal{N}$. When $t=0$, $\mathcal{N}_{0}$ is simply the union of the zero sections of $L_{j}$.

Suppose now the base $D$ has a Calabi-Yau structure $(\omega_{D},\Omega_{D})$, then one can easily write down a $(\mathbb{C}^{*})^{k}$ invariant holomorphic volume form $\Omega_{t}$ on $\mathcal{N}_{t}$ for $t\neq 0$, which is given by

(2.5)

$$\Omega_{t}=\sum_{j=0}^{k}(-1)^{j}\frac{ds_{0}}{s_{0}}\wedge\cdots\widehat{\frac{ds_{j}}{s_{j}}}\wedge\cdots\wedge\frac{ds_{k}}{s_{k}}\wedge\pi_{t}^{*}\Omega_{D},$$

where the notation $\frac{ds_{j}}{s_{j}}(j=0,\cdots k)$ should be understood after choosing a local holomorphic section of $L_{j}$ and it is easy to see that $\Omega_{t}$ does not depend on the particular choice. Also a priori $\Omega_{t}$ is defined away from the singular fibers of the projection $\pi_{t}$, and it is not difficult to see that $\Omega_{t}$ extends to a nowhere vanishing holomorphic volume form on $\mathcal{N}_{t}$.

Let $T^{k}=(S^{1})^{k}\subset(\mathbb{C}^{*})^{k}$ be the obvious maximal compact subgroup. Naturally one would ask for $T^{k}$ invariant Calabi-Yau metrics on (part of) $\mathcal{N}_{t}$ with volume form given by $C\Omega_{t}\wedge\bar{\Omega}_{t}$, and we are then lead to study dimension reduction of the Calabi-Yau equation under the $T^{k}$ action. This has been studied by Matessi [Mat01] and we shall now explain the details for the case $k=1$, and we briefly discuss the case of general $k$ in Section 2.5.

Suppose $(X,\omega,J)$ is an $n$ dimensional Kähler manifold admitting an $S^{1}$ action which is holomorphic and Hamiltonian, with a moment map function $z$, i.e.

(2.6)

$$dz=\xi\lrcorner\omega$$

where $\xi$ is the vector field generating the $S^{1}$ action. We first assume in addition that the $S^{1}$ action is free. Locally in a neighborhood of an $S^{1}$ orbit we can complexify the $S^{1}$ action and obtain a complex quotient $D$ which is an $n-1$ dimensional complex manifold. The local $S^{1}$ quotient can then be identified as a differentiable manifold with $Q=D\times I$, where $I$ is an interval with coordinate function $z$.

Denote by $\{w_{1},\cdots,w_{n-1}\}$ the local holomorphic coordinates on $D$. Then they can be viewed as local holomorphic functions on $X$. Let $t$ be an arbitrary local function with $\xi(t)=1$, Then $\{z,t,w_{1},\cdots,w_{n-1}\}$ gives a local coordinate system on $X$, and we have $\xi=\partial_{t}$. Write $w_{i}=x_{i}+\sqrt{-1}y_{i}$. Then we can express the complex structure $J$ on $X$ in terms of the local coordinates as

(2.7)

$$Jdx_{i}=dy_{i},Jdy_{i}=-dx_{i},Jdz=h^{-1}\Theta,$$

where $h>0$ is a local function and $\Theta$ is a local 1-form which can be written as

(2.8)

$$\Theta=-dt+\theta,$$

such that $\theta$ does not have $dt$ component. The negative sign is due to the fact that

(2.9)

$$Jdz(\partial_{t})=-dz(J\partial_{t})=-\omega(\xi,J\xi)<0.$$

This also gives an intrinsic geometric meaning for $h^{-1}$, as the norm squared of the Killing field $\xi$. In particular $h$ is $S^{1}$ invariant hence descends to a function on $Q$.

By the $S^{1}$ invariance

(2.10)

$$\mathcal{L}_{\xi}(Jdz)=0,\mathcal{L}_{\xi}(dt)=0$$

we obtain

(2.11)

$$\mathcal{L}_{\xi}\theta=0$$

So $\theta$ can also be viewed as a a 1-form on $Q$.

We can write the Kähler form $\omega$ as

(2.12)

$$\omega=dz\wedge(-dt+\theta)+\tilde{\omega}$$

where $\tilde{\omega}$ is a $(1,1)$-form without $dz$ or $dt$ component. This is due to (2.6) and the fact that $\omega$ is of type $(1,1)$. Since $\mathcal{L}_{\xi}\omega=0$ we also have $\mathcal{L}_{\xi}\tilde{\omega}=0$, so the coefficients of $\tilde{\omega}$ also descend to $Q$. In particular, we may view $\tilde{\omega}=\tilde{\omega}(z)$ as a family of $(1,1)$-forms on $D$. The condition $d\omega=0$ is equivalent to

(2.13)

$$\begin{cases}d_{D}\tilde{\omega}(z)=0\\ \partial_{z}\tilde{\omega}(z)=d_{D}\theta,\end{cases}$$

where $d_{D}$ denotes the differential along $D$.

Now we consider the integrability of the complex structure $J$. It is straightforward to check that

(2.14)

$$J\partial_{t}=h^{-1}\theta_{z}\partial_{t}+h^{-1}\partial_{z},$$

so the holomorphic vector field generating the $\mathbb{C}^{*}$ action is given by

(2.15)

$$\xi^{1,0}=\frac{1}{2}(\partial_{t}-\sqrt{-1}J\partial_{t})=\frac{1}{2}(1-\sqrt{-1}h^{-1}\theta_{z})\partial_{t}-\frac{1}{2}\sqrt{-1}h^{-1}\partial_{z}$$

The dual holomorphic $(1,0)$ form is

(2.16)

$$\kappa=\sqrt{-1}(hdz+\sqrt{-1}\Theta+\kappa^{\prime})$$

where $\kappa^{\prime}$ only involves $dx_{i},dy_{i}$. The integrability condition for $J$ can be expressed as

(2.17)

$$d\kappa\wedge\kappa\wedge dw_{1}\wedge\cdots\wedge dw_{n-1}=0.$$

This is then equivalent to

(2.18)

$$\begin{cases}d_{D}\theta\wedge dw_{1}\wedge\cdots\wedge dw_{n-1}=0\\ \partial_{z}\theta=-d_{D}^{c}h\end{cases}$$

where $d_{D}^{c}\equiv J_{D}d_{D}$. The first equation follows from the second equation in (2.13) which implies $d_{D}\Theta$ is of type $(1,1)$ on $D$. Notice (2.13) and (2.18) together can be re-organized as a system

(2.19)

$$\begin{cases}\partial_{z}^{2}\tilde{\omega}+d_{D}d_{D}^{c}h=0\\ d\Theta=\partial_{z}\tilde{\omega}-dz\wedge d_{D}^{c}h\end{cases}$$

It is not difficult to globalize the above discussion and the upshot is that a Kähler metric with a free $S^{1}$ action gives rise to a family of Kähler forms $\tilde{\omega}(z)$ on a complex manifold $D$, together with a positive function $h$ on $D\times I$, satisfying (2.19). This is the familiar procedure in Kähler reduction. The 1-form $-\sqrt{-1}\Theta$ can be viewed as a family of connection 1-forms on the natural $S^{1}$ bundle over $Q$, so as a consequence $\partial_{z}\tilde{\omega}=d_{D}\Theta$ defines an integral cohomology class in $H^{2}(D;\mathbb{Z})$.

Conversely, suppose we are given $\tilde{\omega}(z)$ and $h$ satisfying (2.19), and suppose $[\partial_{z}\tilde{\omega}_{z}]\in 2\pi H^{2}(D;\mathbb{Z})$, then by general theory we can find a connection 1-form $\Theta$ on an $S^{1}$ bundle over $D\times I$ satisfying (2.19), and we can then recover the Kähler metric $(\omega,J)$. Notice there is a possible non-uniqueness caused by the choice of $\Theta$. When $H^{1}(D;\mathbb{R})=0$, different choices of $\Theta$ will differ by an exact 1-form on $D\times I$, so are necessarily gauge equivalent, hence the resulting Kähler metrics will be isomorphic by the induced diffeomorphism.

Now we specialize to Calabi-Yau metrics, so we assume in addition $X$ has a nowhere vanishing holomorphic volume form $\Omega$. Denote the holomorphic $n-1$ form on $X$

(2.20)

$$\tilde{\Omega}=\xi^{1,0}\lrcorner\Omega$$

The fact that $\Omega$ is $S^{1}$ invariant and holomorphic implies that $\tilde{\Omega}$ descends to a holomorphic $(n-1,0)$ form $\Omega_{D}$ on $D$, and we also have

(2.21)

$$\Omega=\kappa\wedge\Omega_{D}.$$

By definition,

(2.22)

$$\omega^{n}=-ndz\wedge dt\wedge\tilde{\omega}^{n-1}$$

and

(2.23)

$$\Omega\wedge\bar{\Omega}=2\sqrt{-1}(-1)^{n-1}hdz\wedge dt\wedge\Omega_{D}\wedge\bar{\Omega}_{D}$$

So the Calabi-Yau equation on $X$

(2.24)

$$\frac{\omega^{n}}{n!}=\frac{(\sqrt{-1})^{n^{2}}}{2^{n}}\Omega\wedge\bar{\Omega}$$

becomes

(2.25)

$$\frac{\tilde{\omega}^{n-1}}{(n-1)!}=\frac{(\sqrt{-1})^{(n-1)^{2}}}{2^{n-1}}h\Omega_{D}\wedge\bar{\Omega}_{D}.$$

Combining (2.19) and (2.25) we get

(2.26)

$$\partial_{z}^{2}\tilde{\omega}+d_{D}d_{D}^{c}\frac{2^{n-1}\tilde{\omega}^{n-1}}{(\sqrt{-1})^{(n-1)^{2}}\Omega_{D}\wedge\bar{\Omega}_{D}}=0.$$

Again it is easy to see this discussion can be globalized so we get a complex Calabi-Yau manifold $(D,\Omega_{D})$ together with a family of Kähler forms $\tilde{\omega}(z)$ satisfying (2.26). Also the converse is true, so the study of $n$ dimensional Calabi-Yau metrics $(X,\omega,\Omega)$ with a free $S^{1}$ action is reduced to the study of the equation (2.26).

Now we make a few observations. First when $n=2$ the equation (2.26) reduces to a linear equation. This is because when $n=2$, $\frac{\sqrt{-1}}{2}\Omega_{D}\wedge\bar{\Omega}_{D}$ is a flat Kähler form and we can write

(2.27)

$$\tilde{\omega}=\frac{\sqrt{-1}}{2}V\Omega_{D}\wedge\bar{\Omega}_{D},$$

for a real function on $Q=D\times I$. Then the equation (2.25) is equivalent to

(2.28)

$$\partial_{z}^{2}V-\Delta_{D}V=0$$

where $\Delta_{D}=d_{D}^{*}d_{D}$ is the Hodge Laplace operator with respect to the above flat metric on $D$. Now (2.28) is exactly the Laplace equation on $Q$, and the above discussion reduces to the classical Gibbons-Hawking ansatz for constructing hyperkähler 4-manifolds. The slight difference is that here we have a distinguished choice of complex structure so the quotient manifold $Q$ naturally splits as $D\times I$.

When $n>2$, (2.26) is still a non-linear equation, and we shall call (2.26) the non-linear Gibbons-Hawking ansatz for Calabi-Yau metrics with $S^{1}$ symmetry. This equation was first written down by Matessi [Mat01].