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].

In general it is not easy to directly solve the equation (2.26), but we can easily see some special solutions, which will be important for us.

Suppose $(D,\Omega)$ is an $n-1$ dimensional compact Calabi-Yau manifold, and $\omega_{D}$ is a Calabi-Yau metric on $D$ with $[\omega_{D}]\in 2\pi H^{2}(D;\mathbb{Z})$, satisfying

(2.29)

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

If we set

(2.30)

$$\begin{cases}\tilde{\omega}(z)=z\cdot\omega_{D};\\ h=z^{n-1}\end{cases}$$

Then as long as $z>0$, $(\tilde{\omega},h)$ clearly satisfy (2.26) and the integrality condition is also achieved, so we get (incomplete) Calabi-Yau metrics in $n$ dimension.

This metric has already appeared in Kähler geometry, which is usually expressed in terms of a Kähler potential. To explain this, we fix a holomorphic line bundle $L_{D}$ with first Chern class $\frac{1}{2\pi}\omega_{D}$, and also fix a hermitian metric on $L_{D}$ whose curvature form is $-\sqrt{-1}\omega_{D}$. Then we consider the subset $\mathcal{C}$ of the total space of $L_{D}$ consisting of all elements $\xi$ with $0<|\xi|<1$. It is endowed with a nowhere vanishing holomorphic volume form $\Omega_{\mathcal{C}}$ and a Ricci-flat Kähler metric $\omega_{\mathcal{C}}$ which is incomplete as $|\xi|\to 1$ and complete as $|\xi|\to 0$. The holomorphic volume form $\Omega_{\mathcal{C}}$ is given by (as in Section 4.2)

(2.31)

$$\Omega_{\mathcal{C}}=\sqrt{-1}\frac{d\xi}{\xi}\wedge\Omega_{D}$$

The metric $\omega_{\mathcal{C}}$ is given by the Calabi ansatz

(2.32)

$$\omega_{\mathcal{C}}=\frac{n}{n+1}\sqrt{-1}\partial\bar{\partial}(-{\log|\xi|^{2}})^{\frac{n+1}{n}}.$$

It is straightforward to check that

(2.33)

$$\omega_{\mathcal{C}}^{n}=\frac{1}{n2^{n-1}}(\sqrt{-1})^{n^{2}}\Omega_{\mathcal{C}}\wedge\overline{\Omega}_{\mathcal{C}},$$

Clearly the Calabi-Yau structure $(\omega_{\mathcal{C}},\Omega_{\mathcal{C}})$ is invariant under the natural $S^{1}$ action on $L_{D}$. Applying the $S^{1}$ reduction as in Section 2.1, we get that the moment map is given by

(2.34)

$$z=(-{\log|\xi|^{2}})^{1/n},$$

and the reduced family of Kähler metrics on $D$ is given by

(2.35)

$$\tilde{\omega}=z\cdot\omega_{D}.$$

The function $h$ is

(2.36)

$$h=\frac{2}{n}z^{n}.$$

So we see this gives rise to the above solution to (2.30) (up to a multiplicative constant on $h$), We call the space $(\mathcal{C},\omega_{\mathcal{C}},\Omega_{\mathcal{C}})$ a Calabi model space. In Section 4.2, Remark 4.12.2 we shall see the formula (2.32) can also be recovered from (2.30), and this works in a more general situation.

Now from the second construction the connection 1-form $\Theta$ is given by the Chern connection 1-form on $L_{D}$. We claim that by varying the holomorphic structures on $L_{D}$ we obtain all the gauge equivalence classes of $\Theta$. This follows from the fact that there is a natural isomorphism between the group $\mathcal{S}_{h}$ of the isomorphism classes of holomorphic line bundles with $c_{1}=0\in H^{2}(D;\mathbb{R})$ and the group $\mathcal{S}_{f}$ of gauge equivalence classes of flat $U(1)$ connections on $D$. Abstractly, we know the first group fits into an exact sequence

(2.37)

$$0\rightarrow\frac{H^{1}(D;\mathcal{O})}{H^{1}(D;\mathbb{Z})}\rightarrow\mathcal{S}_{h}\rightarrow H^{2}_{\text{tor}}\rightarrow 0,$$

where $H^{2}_{tor}$ denotes the torsion subgroup in $H^{2}(D;\mathbb{Z})$, and the second group fits into a short exact sequence

(2.38)

$$0\rightarrow\frac{H^{1}(D;\mathbb{R})}{H^{1}(D;\mathbb{Z})}\rightarrow\mathcal{S}_{f}\rightarrow\text{Hom}(H_{1,\text{tor}},S^{1})\rightarrow 0$$

where $H_{1,tor}$ is the torsion subgroup in $H_{1}(D;\mathbb{Z})$. The isomorphism between $\mathcal{S}_{h}$ and $\mathcal{S}_{f}$ induces an isomorphism on the torsion quotients, which coincides with the isomorphism

(2.39)

$$H^{2}_{\text{tor}}\simeq\text{Ext}(H_{1}(D;\mathbb{Z}),\mathbb{Z})\simeq\text{Hom}(H_{1,tor},S^{1})$$

given by the universal coefficient theorem.

We mentioned in the above that gauge equivalent choices of the connection 1-form $\Theta$ yield isomorphic Calabi-Yau structures on $\mathcal{C}$. Now we observe that for different choices of gauge equivalence classes which differ only by an element in the identity component of $\mathcal{S}_{f}$, the resulting Calabi-Yau structures are also isomorphic, via a diffeomorphism that covers a holomorphic isometry on $D$. For this we fix a choice of $\Theta$, then given any vector field $V$ on $D$, let $\hat{V}$ be the horizontal lift of $V$ to the $U(1)$ bundle with respect to the connection $\Theta$. The infinitesmal variation of $\Theta$ along the flow of $\hat{V}$ is given by

(2.40)

$$\mathcal{L}_{\hat{V}}\Theta=d(\hat{V}\lrcorner\Theta)+\hat{V}\lrcorner d\Theta=V\lrcorner\omega_{D}.$$

Since $\omega_{D}$ is Ricci-flat, every harmonic 1-form on $D$ is parallel, so by Bochner’s theorem, the map $V\mapsto V\lrcorner\omega_{D}$ defines an isomorphism between the space of parallel vector fields on $D$ and the space of harmonic 1-forms on $D$. A parallel vector field is automatically holomorphic and Killing, we see if $\Theta^{\prime}$ differs from $\Theta$ by a harmonic 1-form, then they are related by the flow of some $\hat{V}$ for a parallel vector field $V$.

In the classical Gibbons-Hawking ansatz, to get interesting topology one often needs to allow the $S^{1}$ action to have fixed points. This corresponds to the harmonic function $V$ having Dirac type singularities. For the convenience of later discussion we shall briefly recall the relevant formulae in this model situation, using our description with a preferred complex structure.

We start with $X=\mathbb{C}^{2}$, with standard holomorphic coordinates $(u_{1},u_{2})$, and flat Kähler metric

(2.41)

$$\begin{cases}\omega_{\mathbb{C}^{2}}=\frac{\sqrt{-1}}{2}(du_{1}\wedge d\bar{u}_{1}+du_{2}\wedge d\bar{u}_{2})\\ \Omega_{\mathbb{C}^{2}}=du_{1}\wedge du_{2}.\end{cases}$$

Consider the $S^{1}$ action on $\mathbb{C}^{2}$

(2.42)

$$e^{\sqrt{-1}t}\cdot(u_{1},u_{2})\equiv(e^{-\sqrt{-1}t}u_{1},e^{\sqrt{-1}t}u_{2}).$$

with infinitesimal generator

(2.43)

$$\partial_{t}=-\sqrt{-1}(u_{1}\partial_{u_{1}}-u_{2}\partial_{u_{2}})+\sqrt{-1}(\bar{u}_{1}\partial_{\bar{u}_{1}}-\bar{u}_{2}\partial_{\bar{u}_{2}}).$$

Then we have a moment map $z$ for the $S^{1}$ action with respect to $\omega_{\mathbb{C}^{2}}$ and a complex moment map $y$ for the complexified $\mathbb{C}^{*}$ action with respect to $\Omega_{\mathbb{C}^{2}}$. Together we obtain the standard Hopf map $\pi:\mathbb{C}^{2}\rightarrow Q_{0}\equiv\mathbb{C}\oplus\mathbb{R}$

(2.44)

$$\begin{cases}z=\frac{1}{2}(|u_{1}|^{2}-|u_{2}|^{2})\\ y=u_{1}u_{2}.\end{cases}$$

Then the holomorphic quotient is $D_{0}=\mathbb{C}$ with holomorphic coordinate $y=y_{1}+\sqrt{-1}y_{2}$, and we can calculate that

(2.45)

$$\begin{cases}\tilde{\omega}_{0}=\frac{\sqrt{-1}}{4r}dy\wedge d\bar{y}\\ \Omega_{0}=dy\\ h_{0}=\frac{1}{2r}\\ V=\frac{1}{2r}\end{cases}$$

where $r=\sqrt{y_{1}^{2}+y_{2}^{2}+z^{2}}$ is the standard radial function on $Q_{0}$, and we have the relation

(2.46)

$$r=\frac{1}{2}(|u_{1}|^{2}+|u_{2}|^{2}).$$

The connection 1-form $\Theta_{0}$ on $\mathbb{C}^{2}$ can also be written down explicitly as

(2.47)

$$\Theta_{0}=h\cdot Jdz=\sqrt{-1}\frac{u_{1}d\bar{u}_{1}-\bar{u}_{1}du_{1}+\bar{u}_{2}du_{2}-u_{2}d\bar{u}_{2}}{2(|u_{1}|^{2}+|u_{2}|^{2})}.$$

Define the curvature 2-form on $Q_{0}$

(2.48)

$$\Upsilon_{0}\equiv\partial_{z}\tilde{\omega}_{0}-dz\wedge d_{\mathbb{C}}^{c}h_{0}$$

So we have

(2.49)

$$\Upsilon_{0}=-\frac{\sqrt{-1}}{4r^{3}}(zdy\wedge d\bar{y}+yd\bar{y}\wedge dz-\bar{y}dy\wedge dz)$$

From our above discussion we have the following holds

(2.50)

$$\begin{cases}d\Theta_{0}=\Upsilon_{0}\\ \tilde{\omega}_{0}+dz\wedge\Theta_{0}=\omega_{\mathbb{C}^{2}}\end{cases}$$

where we have implicitly viewed a form on $Q_{0}$ as a form on $\mathbb{C}^{2}$ using the pull-back $\pi^{*}$. In other words, the flat metric on $\mathbb{C}^{2}$ together with the above $S^{1}$ action can be recovered via the Gibbons-Hawking ansatz applied to the function $V=\frac{1}{2r}$ on $Q_{0}=\mathbb{C}\oplus\mathbb{R}$.

Now the above flat metric admits a one-parameter non-flat perturbation, corresponding to replacing $V$ by $V+T$ for a positive constant $T$. Correspondingly we have

(2.51)

$$\begin{cases}\tilde{\omega}_{0,T}=(\frac{1}{2r}+T)\frac{\sqrt{-1}}{2}dy\wedge d\bar{y}\\ h_{0,T}=\frac{1}{2r}+T\\ \end{cases}$$

This yields a family of Taub-NUT metrics $(\omega_{TN,T},\Omega_{TN,T})$ on $\mathbb{R}^{4}$ with

(2.52)

$$\begin{cases}\omega_{TN,T}\equiv(\frac{1}{2r}+T)\frac{\sqrt{-1}}{2}dy\wedge d\bar{y}+dz\wedge\Theta_{0}\\ \Omega_{TN,T}\equiv\sqrt{-1}((\frac{1}{2r}+T)dz+\Theta_{0})\wedge dy.\end{cases}$$

We can still view these metrics as defined on $\mathbb{R}^{4}$ with coordinates $u_{1},u_{2},\bar{u}_{1},\bar{u}_{2}$ via the above Hopf map, but the coordinate functions $u_{1},u_{2}$ are no longer holomorphic. Indeed one can write down explicitly the holomorphic volume form

(2.53)

$$\Omega_{TN,T}=du_{1}\wedge du_{2}+\frac{T}{2}dz\wedge dy.$$

We also have

(2.54)

$$h_{0,T}dz+\sqrt{-1}\Theta_{0}=Tdz+\frac{1}{2}(\frac{du_{1}}{u_{1}}-\frac{du_{2}}{u_{2}}).$$

LeBrun [LeB91] showed that if we make a (non-holomorphic) coordinate change on $\mathbb{C}^{2}$

(2.55)

$$\begin{cases}\eta_{+}=u_{1}e^{\frac{T}{2}(|u_{1}|^{2}-|u_{2}|^{2})}\\ \eta_{-}=u_{2}e^{\frac{T}{2}(|u_{2}|^{2}-|u_{1}|^{2})}\end{cases}$$

then we have

(2.56)

$$\Omega_{TN,T}=d\eta_{+}\wedge d\eta_{-}.$$

So that the underlying complex manifold is still bi-holomorphic to $\mathbb{C}^{2}$ with holomorphic coordinates $\eta_{+}$ and $\eta_{-}$, and one can write down a global Kähler potential

(2.57)

$$\begin{cases}\omega_{TN,T}=\sqrt{-1}\partial\bar{\partial}\varphi_{T},\\ \varphi_{T}=\frac{1}{2}(|u_{1}|^{2}+|u_{2}|^{2})+\frac{T}{4}(|u_{1}|^{4}+|u_{2}|^{4}).\end{cases}$$

Again in Section 4.2, Remark 4.12.3 we shall see this follows from a more general fact.

Now we return to the higher dimensional situation. One interesting type of singularities is locally modeled on the product of the above 2 dimensional model with a flat space $\mathbb{C}^{n-3}$. Then the space $Q\equiv\mathbb{C}^{n-3}\oplus Q_{0}$ is still smooth and the singular set of $\tilde{\omega}$ and $h$ is the real codimension 3 subspace $P=\mathbb{C}^{n-1}\oplus\{0\}\subset Q$, and they both have transversal Dirac type singularities along $P$. In our applications, we need to consider the non-linear situation. So $D$ is an $n-1$ dimensional complex manifold and $H\subset D$ is a smooth complex hypersurface, and we want our solution $(\tilde{\omega},h)$ to the equation (2.26) to satisfy a distributional equation on $Q$ of the form

(2.58)

$$(\partial_{z}^{2}\tilde{\omega}+d_{D}d_{D}^{c}\frac{2^{n-1}\tilde{\omega}^{n-1}}{(\sqrt{-1})^{(n-1)^{2}}\Omega_{D}\wedge\bar{\Omega}_{D}})\wedge dz=2\pi\cdot\delta_{P}$$

where $P\equiv H\times\{0\}$ and $\delta_{P}$ is a degree $3$ current given by integration along $P$. This equation has appeared in the literature [Zha04] in a slightly different form. A solution to this equation, with suitable regularity, will give rise to a Calabi-Yau metric with an $S^{1}$ action whose fixed point locus is a complex codimension two submanifold and transverse to which the action is modeled on the above standard $S^{1}$ action on $\mathbb{C}^{2}$. This is exactly what we are motivated to search for from the algebro-geometric discussion at the beginning of this section.

Unfortunately, solving the non-linear equation together with distribution (2.58) in general seems very difficult. Motivated by recent results in the study of adiabatic limits of $G_{2}$ manifolds [Don17, FHN17], we attempt to study the equation when the $S^{1}$ orbit is very small. Again suppose $(D,\omega_{D},\Omega_{D})$ is $n-1$ dimensional Calabi-Yau, then for $T$ large we know there are trivial constant solutions with $\tilde{\omega}=T\omega_{D}$ and $h=T^{n-1}$. Now we look for a perturbation $\tilde{\omega}=T\omega+\psi$ for $T$ large. To the first order we know $\psi$ must satisfy the linearized equation at $T\omega$, hence

(2.59)

$$\partial_{z}^{2}\psi+T^{n-2}d_{D}d_{D}^{c}\Tr_{\omega_{D}}\psi=0,$$

which by Kähler identities is equivalent to

(2.60)

$$\partial_{z}^{2}\psi-T^{n-2}d_{D}d_{D}^{*}\psi=0.$$

Up to a scaling of the $z$ variable this is equivalent to the equation

(2.61)

$$\partial_{z}^{2}\psi-d_{D}d_{D}^{*}\psi=0.$$

If we can at the same time achieve $d_{D}\psi=0$, then this is equivalent to that $\psi\wedge dz$ being a harmonic 3-form on the product $Q=D\times\mathbb{R}_{z}$. Again the interesting case is when $\psi$ has singularities, and we want to study the case when the singular set is of the form $H\times\{0\}\subset Q$ for $H$ a smooth hypersurface in $D$, and correspondingly $\psi$ satisfies

(2.62)

$$\Delta_{Q}\psi=2\pi\cdot\delta_{P}$$

This is a generalization of Green’s function to 3-forms and we shall call it a Green’s current, which is our main object of study in Section 3.

When $n=2$, the above Green’s current is simply the Green’s function and this has been used in [HSVZ18] to obtain exact solutions to a family of incomplete Calabi-Yau metric by Gibbons-Hawking construction. In higher dimension using Green’s current we can apply (2.19) to define a family of approximately Calabi-Yau metrics. This is our main object of study in Section 4.

For our geometric application in this paper the family of incomplete approximately Calabi-Yau metrics will be sufficient, see Section 7. On the other hand, one can also perturb these to genuine Calabi-Yau metrics. This will be discussed in Section 6.

Now we assume an $n$ dimensional Kähler manifolds $(X,\omega,J)$ admits an $T^{k}(k\geq 1)$ action which is holomorphic and Hamiltonian. We first assume the action is free. Let $(z_{1},\cdots,z_{k})$ be the moment map. Then similar discussion to that in Section 2.1 yields locally a family of Kähler forms $\tilde{\omega}$ on the complex quotient, parametrized by $(z_{1},\ldots z_{k})\in\mathbb{R}^{k}$, a family of connection $1$-forms $-\sqrt{-1}\Theta_{j}(j=1,\cdots,k)$ and a positive definite $k\times k$ real symmetric matrix $W=(W_{ij})$ with the inverse matrix

(2.63)

$$W^{ij}=\langle\partial_{t_{i}},\partial_{t_{j}}\rangle,$$

such that the following system of equations hold

(2.64)

$$\begin{cases}\partial_{z_{j}}\tilde{\omega}=d_{D}\Theta_{j}\\ \partial_{z_{j}}\Theta_{i}=-d^{c}_{D}W_{ij}\\ \partial_{z_{l}}W_{ij}=\partial_{z_{j}}W_{il}.\end{cases}$$

As before the first two equations combine to give an equation on $(\tilde{\omega},W^{ij})$

(2.65)

$$\partial_{z_{i}}\partial_{z_{j}}\tilde{\omega}+d_{D}d_{D}^{c}W_{ij}=0.$$

Now suppose the complex quotient $D$ is Calabi-Yau with a holomorphic volume form $\Omega_{D}$, then the Calabi-Yau equation on $X$ becomes

(2.66)

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

This equation has been derived by Matessi [Mat01] and Zharkov [Zha04]. Again when the $T^{k}$ action is not free one should replace (2.65) by a distributional equation. We will discuss a simplest example in Section 8.1. In the most extreme case when $k=n$ is the complex dimension of $X$, this becomes the real Monge-Ampère equation

(2.67)

$$\det(W_{ij})=C.$$