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.

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:

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.