Let $n\geq 2$ be a positive integer and $\Delta$ be the unit disc in $\mathbb{C}$. Let $p:(\mathcal{X},\mathcal{L})\rightarrow\Delta$ be a flat polarized degenerating family of $n$-dimensional Calabi-Yau varieties. More precisely, we assume that $\mathcal{X}$ is normal with $X_{t}\equiv p^{-1}(t)$ smooth for $t\neq 0$ and $X_{0}$ singular, the relatively canonical line bundle $K_{\mathcal{X}/\Delta}$ is trivial, and $\mathcal{L}$ is relatively ample. Yau’s proof of the Calabi conjecture [Yau78] yields for each $t\neq 0$ a unique smooth Ricci-flat Kähler metric (the Calabi-Yau metric) $\omega_{CY,t}$ on $X_{t}$ in the cohomology class $2\pi c_{1}(\mathcal{L}|_{X_{t}})$. A folklore problem is to understand the limiting geometric behavior of these metrics in the Gromov-Hausdorff sense, as $t$ tends to zero, and the connection with the algebraic geometry associated to this degeneration. A particularly intriguing and challenging situation is when collapsing occurs, i.e. when the diameters of $\omega_{CY,t}$ tend to infinity, and if we rescale the diameters to be fixed then the Gromov-Hausdorff limit is a lower dimensional space.

Our goal in this paper is to study one special class of complex structure degenerations, and give a relatively complete description of the collapsing geometry of the family of Calabi-Yau metrics. We shall mainly focus on the example below, but the crucial techniques involved apply to more abstract situation, and the strategy can possibly extend to more general classes of collapsing, see the discussions in Section 8.

Let $f_{1},f_{2},f$ be homogeneous polynomials in $n+2$ variables of degree $d_{1}$, $d_{2}$ and $d_{1}+d_{2}=n+2$ respectively. Let $\mathcal{X}\subset\mathbb{C}\mathbb{P}^{n+1}\times\Delta$ be the family of Calabi-Yau hypersurfaces in $\mathbb{C}\mathbb{P}^{n+1}$ defined by the equation $F_{t}(x)=0$ (see Figure 1.1), where

(1.1)

$$F_{t}(x)\equiv f_{1}(x)f_{2}(x)+tf(x),$$

and $t$ is the complex parameter on the unit disc $\Delta\subset\mathbb{C}$. The relative ample line bundle $\mathcal{L}$ comes from the natural $\mathcal{O}(1)$ bundle over $\mathbb{C}\mathbb{P}^{n+1}$.

We further assume $f_{1},f_{2},f$ are sufficiently general so that the following hold:

1. (i)

   $X_{0}=Y_{1}\cup Y_{2}$, where $Y_{1}=\{f_{1}(x)=0\}$ and $Y_{2}=\{f_{2}(x)=0\}$ are smooth hypersurfaces in $\mathbb{C}\mathbb{P}^{n+1}$;

2. (ii)

   $X_{t}$ is smooth for $t\neq 0$.;

3. (iii)

   $D=\{f_{1}(x)=f_{2}(x)=0\}$ is a smooth complete intersection in $\mathbb{C}\mathbb{P}^{n+1}$;

4. (iv)

   $H=\{f_{1}(x)=f_{2}(x)=f(x)=0\}$ is a smooth complete intersection in $\mathbb{C}\mathbb{P}^{n+1}$.

For $i=1,2$, it has been shown by Tian-Yau [TY90] that $Y_{i}\setminus D$ admits a complete Ricci-flat Kähler metric $\omega_{TY,i}$ with interesting asymptotics governed by the Calabi model space (c.f. Section 2.2) , and the latter in turn depends on the Calabi-Yau metric on $D$ in the cohomology class $2\pi\mathcal{O}(1)|_{D}$. The properties of Tian-Yau metrics will be briefly reviewed in Section 7.2. Here we point out that the construction of Tian-Yau can be viewed as a generalization of Yau’s proof of Calabi conjecture to the non-compact case, but it is not clear in what sense the metrics $\omega_{TY,i}$ are uniquely or canonically associated to the pair $(Y_{i},D)$ in suitable sense.

The main result of this paper is as follows:

Indeed, using gluing construction in this paper we give a fairly precise description of the multi-scale collapsing of the Calabi-Yau metrics $(X_{t},\omega_{CY,t})$ as $t\rightarrow 0$. One consequence is that the Tian-Yau metrics, though not a priori canonical by construction, is indeed canonically associated to the degenerations of compact Calabi-Yau manifolds.

We also remark that when $n=2$ a gluing construction for hyperkähler metrics is done in [HSVZ18], and Figure 1.2 is essentially the same as Figure 1.1 in [HSVZ18]. But there are several different features

• •

  When $n=2$, the construction in [HSVZ18] is more general, in the sense that given any two Tian-Yau metrics in complex two dimension, we can construct a neck region connecting them together and then perturb to genuine hyperkähler metrics. In higher dimensions a general gluing construction at the level of Calabi-Yau metrics (without a priori knowing the complex structures) seems lacking. We leave this for future exploration and see Section 8.2 for a discussion from the technical viewpoint.

• •

  One of the motivation for the work [HSVZ18] was the problem we solved in this paper, however in [HSVZ18] we were only able to perform the gluing construction at the level of hyperkähler metrics and the information on complex structure was lost. It has been an interesting question to understand the complex geometric meaning of the construction in [HSVZ18]. There are possible approaches, by appealing to the period mapping and the global Torelli theorem, to recover the complex structures abstractly. In this paper however, we directly work on the complex family, and are able to draw direct connection to complex geometry for all dimensions. From a technical point of view we are in a more rigid situation, and we need to perform analysis at the level of Kähler potentials.

• •

  A crucially new technical difficulty in higher dimensions is related to the construction of the neck region. When $n=2$, we used the linear Gibbons-Hawking ansatz which gives exactly the $S^{1}$ invariant incomplete hyperkähler transition region. When $n>2$ the $S^{1}$ reduction of the Calabi-Yau equation (what we call the non-linear Gibbons-Hawking ansatz, see Section 2) is no longer linear, and it seems difficult to solve the non-linear equation directly. Instead we shall only use a singular solution to the linearized equation to construct approximately Calabi-Yau metrics. This suffices for the gluing argument. A substantial amount of analysis in this paper is required to deal with this singular solution, and the corresponding singular geometry.

The proof of Theorem 1.1 consists of roughly three main pieces.

The first piece involves algebraic modification of the family $\mathcal{X}$. Our initial naive strategy is to start with the Tian-Yau metrics on $X_{0}\setminus D=(Y_{1}\setminus D)\cup(Y_{2}\setminus D)$, and graft them to nearby fibers $X_{t}$ for $|t|$ small to get Kähler metrics which are approximately Calabi-Yau. However, the existence of singularities of the total space $\mathcal{X}$ along $H$ imposes difficulties in performing a reasonable construction. So our first step is to modify the family $\mathcal{X}\rightarrow\Delta$ to another family $\widehat{\mathcal{X}}\rightarrow\Delta$ using base change and birational modifications (c.f. Figure 7.1). The new family $\widehat{\mathcal{X}}$ agrees with $\mathcal{X}$ away from $X_{0}$, and the new fiber $\widehat{X}_{0}$ consists of a chain of three components, with the two end components isomorphic to $Y_{1},Y_{2}$ respectively, and the middle component $\mathcal{N}$ is given by a conic bundle over $D$, as a natural hypersurface in the projective bundle $\mathbb{P}(L_{1}\oplus L_{2}\oplus\mathbb{C})$ cut out by the equation $s_{1}s_{2}+s_{3}f(x)=0$. The family of conics degenerate precisely along the divisor $H$ in $D$. The component $\mathcal{N}$ intersects transversally with $Y_{1},Y_{2}$ along $D_{1},D_{2}$, which are naturally isomorphic to $D$. Notice $\widehat{\mathcal{X}}$ is not necessarily smooth. Indeed it has singularities along $D_{1}\cup D_{2}$ which is of codimension two. However it turns out that working with $\widehat{\mathcal{X}}$ is the correct thing to do. This is done in Section 7.1.

The second piece involves the construction of the neck region. We want Calabi-Yau metrics on the smooth locus of the central fiber of $\widehat{\mathcal{X}}$. For the two end components these are provided by the complete Tian-Yau metrics. For the middle component, with a moments’ thought one realizes that it is difficult to construct a complete Calabi-Yau metric on $\mathcal{N}^{0}=\mathcal{N}\setminus(D_{1}\cup D_{2})$. The reason is that if such metric existed, it would have two ends, and Ricci-flatness would imply it must split a line, and this is not quite compatible with the complex geometry of $\mathcal{N}^{0}$. Instead we shall look for a family of incomplete Calabi-Yau metrics defined on larger and larger open subsets in $\mathcal{N}^{0}$. The fact that $\mathcal{N}$ has a natural holomorphic $\mathbb{C}^{*}$ action suggests us to look for Calabi-Yau metrics with $S^{1}$ symmetry.

In complex dimension 2, this is essentially achieved in [HSVZ18] using the classical Gibbons-Hawking ansatz (except we did not identify the underlying complex manifold). In higher dimensions the technical details are more complicated. In Section 2 we discuss a higher dimensional generalization of the Gibbons-Hawking ansatz. The corresponding reduced equation is still non-linear, and by linearization we are lead to study certain solutions to a linear elliptic PDE with singularities along a submanifold. The existence and local regularity of such solutions, which we call Green’s currents, is studied in detail in Section 3. In Section 4 we use these Green’s currents to construct a family of incomplete Kähler metrics on open subsets of $\mathcal{N}^{0}$. The fact that the singularities of the Green’s currents are non-isolated causes difficulties in understanding the regularity of the Kähler metrics. In reality we only prove the metrics are $C^{2,\alpha}$ and this suffices for our purpose. Another difference in higher dimensions is that these metrics are only approximately Calabi-Yau. In Section 4 we study the various rescaled limit geometries for this family of metrics. We also give a formula for the Kähler potential of these Kähler metrics, which is crucial for our gluing construction since we work on the fixed complex family $\widehat{\mathcal{X}}$. In Section 7.3 we graft the incomplete Calabi-Yau metrics constructed in Section 4 and the complete Tian-Yau metrics on $Y_{i}\setminus D$ to $C^{1,\alpha}$ Kähler metrics on $X_{t}$ for $|t|$ sufficiently small, which are approximately Calabi-Yau.

The third piece then involves weighted analysis. This is roughly along the same lines as in [HSVZ18]. Again a new difficult point is the proof of a Liouville theorem on the Tian-Yau spaces. This will be done in Section 5 using elementary analysis of special functions. For readers’ convenience, we also summarize the relevant formulae regarding these special functions in Appendix A. In Section 6 we use the implicit function theorem and weighted estimates to show the family of approximately Calabi-Yau metrics on the neck can be perturbed to genuine Calabi-Yau metrics. Here a subtle point is that we use Neumann boundary condition instead of Dirichlet boundary condition. One can then see directly from this the Gromov-Hausdorff collapsing behavior of the Calabi-Yau metrics. We also discuss the renormalized limit measures.

Notice for the proof of Theorem 1.1 we do not need to use these incomplete Calabi-Yau metrics, but the proof will involve similar arguments. This is explained in Section 7.4.

We would like to thank Lorenzo Foscolo, Mark Haskins, and Shouhei Honda for helpful discussions. We are also grateful to Hans-Joachim Hein and Jeff Viaclovsky for the stimulating discussions on the study of collapsing hyperkähler metrics on K3 surfaces which led to an earlier joint paper [HSVZ18]. We thank Yang Li for communications regarding the draft of his preprint [Li19] and the rough draft of the current paper in January 2019. Substantial parts of this paper were written when the second author was visiting Princeton University, Sinica Academia and ShanghaiTech University in the academic year 2018-2019. He would like to acknowledge the hospitality and support of those institutions.

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 $Q_{0}=\mathbb{C}\oplus\mathbb{R}$.