1.1. Background and main results [04YY]

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1.1. Background and main results

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:

(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 $\mathbb{C}\mathbb{P}^{n+1}$ ;
(ii)

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

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

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

In particular by adjunction formula $D$ is also Calabi-Yau, and we may identify $D$ with $D\times\{0\}$ which sits as an anti-canonical divisor in both $Y_{1}$ and $Y_{2}$ . Moreover, the normal bundle $L_{i}$ of $D$ in $Y_{i}$ is given by $\mathcal{O}(d_{3-i})|_{D}$ . Notice the total space $\mathcal{X}$ has singularities along $H\times\{0\}$ , and transverse to $H\times\{0\}$ the singularities are locally modeled on a three dimensional ordinary double point. The dual intersection complex of the singular fiber $X_{0}$ is a one dimensional interval.

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.

Figure 1.1. The algebraic family

\mathcal{X}

The main result of this paper is as follows:

Theorem 1.1.

For $|t|$ sufficiently small, the Calabi-Yau metrics $\omega_{CY,t}$ on $X_{t}$ can be constructed by gluing the Tian-Yau metrics on $Y_{1}\setminus D$ and $Y_{2}\setminus D$ , together with an approximately Calabi-Yau metric on a transition region. We normalize $\omega_{CY,t}$ so that $\diam(X_{t},{\omega_{CY,t}})=1$ , and we denote by $d\underline{\nu}_{t}$ the renormalized measure of $(X_{t},\omega_{CY,t})$ . Then the following holds as $t\to 0$ (see Figure 1.2):

(1)

Under measured Gromov-Hausdorff convergence, the spaces collapse to the unit interval with a singular renormalized limit measure, that is,

(1.2)

(X_{t},\omega_{CY,t},d\underline{\nu}_{t})\xrightarrow{mGH}(\mathbb{I},dv^{2},d\underline{\nu}_{\mathbb{I}}),

where $\mathbb{I}=[-\frac{d_{1}}{d_{1}+d_{2}},\frac{d_{2}}{d_{1}+d_{2}}]\subset\mathbb{R}$ is a unit interval with the standard metric $dv^{2}$ and

(1.3)		$\displaystyle d\underline{\nu}_{\mathbb{I}}$	$\displaystyle=C_{n,d_{1},d_{2}}\cdot\mathscr{V}_{\mathbb{I}}(v)dv,$
(1.4)		$\displaystyle\mathscr{V}_{\mathbb{I}}(v)$	$\displaystyle=\begin{cases}(\frac{v}{d_{1}}+\frac{1}{d_{1}+d_{2}})^{\frac{n-1}{n+1}},&v\in[-\frac{d_{1}}{d_{1}+d_{2}},0],\\ (-\frac{v}{d_{2}}+\frac{1}{d_{1}+d_{2}})^{\frac{n-1}{n+1}},&v\in[0,\frac{d_{2}}{d_{1}+d_{2}}],\end{cases}$

for some constant $C_{n,d_{1},d_{2}}>0$ .

(2)

There is a continuous surjective map $\mathcal{F}_{t}:X_{t}\rightarrow\mathbb{I}$ with the following properties:

(a)

(Almost distance preserving) For all $p,q\in X_{t}$ ,

(1.5)

\Big||\mathcal{F}_{t}(p)-\mathcal{F}_{t}(q)|-d_{\omega_{CY,t}}(p,q)\Big|<\tau(t),\quad\lim\limits_{t\to 0}\tau(t)=0.

(b)

(Regular fiber) For each $v\in(-\frac{d_{1}}{d_{1}+d_{2}},\frac{d_{2}}{d_{1}+d_{2}})\setminus\{0\}$ , the fiber $\mathcal{F}_{t}^{-1}(v)$ is an $S^{1}$ -fiber bundle $S^{1}\to\mathcal{F}_{t}^{-1}(v)\xrightarrow{\pr_{v}}D$ with the first Chern class

(1.6)

\displaystyle c_{1}(\mathcal{F}_{t}^{-1}(v))=\begin{cases}c_{1}(\mathcal{O}(d_{2})|_{D}),&v\in(-\frac{d_{1}}{d_{1}+d_{2}},0),\\ c_{1}(\mathcal{O}(-d_{1})|_{D}),&v\in(0,\frac{d_{2}}{d_{1}+d_{2}}).\end{cases}

(c)

(Singular fiber and deepest bubble) The fiber $\mathcal{F}_{t}^{-1}(0)$ is a singular $S^{1}$ -fibration over $D$ with vanishing circles along $H\subset D$ . Suitable rescalings around the vanishing circles on $\mathcal{F}_{t}^{-1}(0)$ converge to the Riemann product $\mathbb{C}_{TN}^{2}\times\mathbb{C}^{n-2}$ , where $\mathbb{C}_{TN}^{2}$ is the Taub-NUT space.
(d)

(End bubble) Suitable rescalings around the ends $v_{-}=-\frac{d_{1}}{d_{1}+d_{2}}$ and $v_{+}=\frac{d_{2}}{d_{1}+d_{2}}$ converge to the complete Tian-Yau metrics $\omega_{TY,1}$ and $\omega_{TY,2}$ on $Y_{1}\setminus D$ and $Y_{2}\setminus D$ respectively.

Remark 1.1.1.

The transition region with approximately Calabi-Yau metrics Theorem 1.1 is constructed in Section 4, which is the main geometric input of this paper. From the geometric point of view, for $|t|\ll 1$ , the transition region is approximately determined by the geometry of the Calabi-Yau metric on $D$ , hence reduces to one lower dimension, and yet it occupies most of the volume of $X_{t}$ . On the other hand, interesting topologies in $X_{t}$ are located in the two Tian-Yau ends, which have relatively small volume.

Remark 1.1.2.

In Theorem 1.1, the singularity of $d\underline{\nu}_{\infty}$ occurs at $v=0\in\mathbb{I}$ which coincides with the locus of the singular fiber $\mathcal{F}^{-1}(0)$ . The explicit formula of $d\underline{\nu}_{\infty}$ has applications in understanding the properties of general collapsing limits with lower Ricci curvature bound. See Section 6.4 for more discussions.

Remark 1.1.3.

The fibration $\mathcal{F}_{t}$ has a multi-scale collapsing nature in the following sense: each fiber $\mathcal{F}_{t}^{-1}(v)$ with $v\in(-\frac{d_{1}}{d_{1}+d_{2}},\frac{d_{2}}{d_{1}+d_{2}})\subset\mathbb{I}$ has a further $S^{1}$ -collapsing direction given by the fibration

S^{1}\to\mathcal{F}_{t}^{-1}(v)\xrightarrow{\pr_{v}}D

such that, as $t\to 0$ , the collapsing rate of the $S^{1}$ fibers is of higher order than $\mathcal{F}_{t}^{-1}(v)$ . This iterated collapsing in effect gives rise to various bubbles of different geometric natures. See Section 4.3 for detailed studies on the rescaled limit geometries.

Remark 1.1.4.

Transverse to the divisor $H\subset D$ , the singular fibration $\mathcal{F}_{t}$ is topologically modeled on the composition of the Hopf fibration

\mathbb{C}^{2}\rightarrow\mathbb{C}\oplus\mathbb{R}\simeq\mathbb{R}^{3};\ (z_{1},z_{2})\mapsto(z_{1}z_{2},\frac{1}{2}(|z_{1}|^{2}-|z_{2}|^{2})).

and the projection map

\mathbb{C}\oplus\mathbb{R}\rightarrow\mathbb{R};(y,z)\mapsto z.

Figure 1.2. The collapsing Calabi-Yau metric

\omega_{CY,t}

X_{t}

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.