2. Preliminaries [05DA]

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2. Preliminaries

In this section, we review some notions and results, which will be used in the proof of Theorem 1.1.

2.1. Cheeger-Gromov convergence

Since Gromov introduced the concept of Gromov-Hausdorff topology in [14], the convergence of Riemannian manifolds was studied from various perspectives (c.f. [1], [2], [7], [8], [11], [13], [19], [35], [38] and references in [9]). In [14] and [13], a convergence theorem, the Cheeger-Gromov convergence theorem, was proved for Riemannian manifolds with bounded curvature and non-collapsing. The Kähler version of this theorem can be found in [30]. See [7] for the convergence of manifolds with other holonomy groups.

Theorem 2.1 (Kähler version of Cheeger-Gromov convergence theorem).

Let $\{(M_{k},g_{k},J_{k},\omega_{k},p_{k})\}$ be a family of pointed compact Kähler n-manifolds with sectional curvature and injectivity radius at $p_{k}$

|K_{g_{k}}|\leq 1,\ \ \ i_{g_{k}}(p_{k})\geq C,

for a constant $C>0$ independent of $k$ . Then a subsequence of $\{(M_{k},g_{k},J_{k},\omega_{k},p_{k})\}$ converges to a complete Kähler n-manifold $(X,g,J,\omega,p)$ in the pointed $C^{1,\alpha}$ -sense, i.e. for any $r>0$ , there are embeddings $F_{k,r}:B_{g}(p,r)\longrightarrow M_{k}$ such that $F_{k,r}(p)=p_{k}$ , $F_{k,r}^{*}g_{k}$ (resp. $dF_{k,r}^{-1}J_{k}dF_{k,r}$ and $F_{k,r}^{*}\omega_{k}$ ) converges to $g$ (resp. $J$ and $\omega$ ) in the $C^{1,\alpha}$ -sense.

If we assume that $g_{k}$ are Einstein metrics, it is shown in [1] that, by passing to a subsequence, $\{(M_{k},g_{k},J_{k},\omega_{k},p_{k})\}$ converges to $(X,g,J,\omega,p)$ in the pointed $C^{\infty}$ -sense, and $g$ is also an Einstein metric, i.e. $F_{k,r}^{*}g_{k}$ (resp. $dF_{k,r}^{-1}J_{k}dF_{k,r}$ and $F_{k,r}^{*}\omega_{k}$ ) converges to $g$ (resp. $J$ and $\omega$ ) in the $C^{\infty}$ -sense. Assume that $(M_{k},g_{k},J_{k},\omega_{k},p_{k})$ are Ricci-flat Calabi-Yau manifolds, and $\Omega_{k}$ are the corresponding holomorphic volume forms. Since $\Omega_{k}$ are parallel, i.e. $\nabla^{g_{k}}\Omega_{k}\equiv 0$ , for any $r>0$ , $F_{k,r}^{*}\Omega_{k}$ converge to a holomorphic volume form $\Omega$ on $X$ in the $C^{\infty}$ -sense, and $(X,g,J,\omega,\Omega)$ is a complete Ricci-flat Calabi-Yau $n$ -manifold.

In [5], [6], the collapsing of Riemannian manifolds with bounded curvature was studied by combining blow-up arguments and the Cheeger-Gromov convergence theorem. It was shown that there is a constant $\epsilon_{0}(n)>0$ depending only on $n$ such that there is an $F$ -structure $\mathcal{F}$ of positive rank on a region covering $M_{\epsilon_{0}}$ in a Riemannian $n$ -manifold $(M,g)$ , where $M_{\epsilon_{0}}$ denotes the subset with injectivity radius $i_{g}(p)<\epsilon_{0}$ and sectional curvature $\sup\limits_{B_{g}(p,1)}|K_{g}|\leq 1,$ for any $p\in M_{\epsilon_{0}}$ . See [5] and [6] for the definition of $F$ -structure of positive rank. If we assume that $g$ is a Kähler metric, some additional information about the $F$ -structure $\mathcal{F}$ is expected. We have the following conjecture:

Conjecture 2.2.

For any $n\in\mathbb{N}$ , there exists a constant $\epsilon=\epsilon(n)>0$ depending only on $n$ such that, if $(M,\omega,J,g)$ is a closed Kähler n-manifold with $[\omega]\in H^{2}(M,\mathbb{Z})$ , and

M_{\epsilon}=\{p\in M|\ i_{g}(p)<\epsilon,\ \sup_{B_{g}(p,1)}|K_{g}|\leq 1\},

then there is an open subset $W\subset M$ such that $W\supset M_{\epsilon}$ , and $W$ admits an F-structure $\mathcal{F}$ of positive rank, whose orbits $\mathcal{O}_{p}$ , $p\in M_{\epsilon}$ , are isotropic submanifolds of $(M,\omega)$ , i.e.

\omega|_{\mathcal{O}_{p}}\equiv 0.

We will address this question in other papers. In the present paper, we prove Theorem 1.1 by combining Theorem 2.1 and the deformation theory of special lagrangian fibrations.

2.2. Implicit function theorem

For studying the deformation of special lagrangian fibrations, we need the following quantity version of implicit function theorem.

Theorem 2.3 (Theorem 3.2 in [31]).

Let $(\mathfrak{B}_{1},\|\cdot\|_{1})$ and $(\mathfrak{B}_{2},\|\cdot\|_{2})$ be two Banach spaces, $\|\cdot\|_{E}$ be the standard Euclidean metric on $\mathbb{R}^{n}$ , $U\subset\mathbb{R}^{n}\times\mathfrak{B}_{1}$ be an open set, and $\mathfrak{F}:U\longrightarrow\mathfrak{B}_{2}$ be a continuously differentiable map. Denote the differential

D\mathfrak{F}(y,\sigma)(\dot{y}+\dot{\sigma})=D_{y}\mathfrak{F}(y,\sigma)\dot{y}+D_{\sigma}\mathfrak{F}(y,\sigma)\dot{\sigma},

for $(y,\sigma)\in U$ , $\dot{y}\in\mathbb{R}^{n}$ and $\dot{\sigma}\in\mathfrak{B}_{1}$ . Assume that $(0,0)\in U$ satisfies that $D_{\sigma}\mathfrak{F}(0,0):\mathfrak{B}_{1}\longrightarrow\mathfrak{B}_{2}$ has a bounded linear inverse $D_{\sigma}\mathfrak{F}(0,0)^{-1}:\mathfrak{B}_{2}\longrightarrow\mathfrak{B}_{1}$ with

\|D_{\sigma}\mathfrak{F}(0,0)^{-1}\|\leq\overline{C}

for a constant $\overline{C}>0$ . Let $r>0$ , $\delta_{0}>\delta>0$ be constants such that, if $\|y_{0}\|_{E}<r,$ and $\|\sigma\|_{1}\leq\delta_{0}$ , then $(y_{0},\sigma)\in U$ , and

\|D_{\sigma}\mathfrak{F}(y_{0},\sigma)-D_{\sigma}\mathfrak{F}(0,0)\|\leq\frac{1}{2\overline{C}}\ \ {\rm and}\ \ \|\mathfrak{F}(y_{0},0)\|_{2}\leq\frac{\delta}{4\overline{C}}.

Then, for any $\|y\|_{E}<r$ , there exists a unique $\sigma(y)\in\mathfrak{B}_{1}$ such that

\mathfrak{F}(y,\sigma(y))=0,\ \ \ \|\sigma(y)\|_{1}\leq\delta.

Furthermore,

D\sigma(y)\dot{y}=-D_{\sigma}\mathfrak{F}(y,\sigma)^{-1}D_{y}\mathfrak{F}(y,\sigma)\dot{y}.

The difference between this version of implicit function theorem and the usual one (c.f. [20]) is that we use the condition $\|\mathfrak{F}(y,0)\|_{2}\leq\frac{\delta}{4\overline{C}}$ to replace the condition $\mathfrak{F}(0,0)=0$ besides other quantity estimates.