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

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.