We recall that a Calabi-Yau metric on a complex manifold $X$ is a Kähler metric with vanishing Ricci curvature. If such a metric exists then $c_{1}({TX})=0\in H^{2}(X,{{\bf R}})$ and hence the class of the canonical bundle $\bigwedge^{\dim X}(T^{*}X)$ is torsion in $Pic(X)$. According to the famous Yau theorem, for any compact Kähler manifold $X$ such that $c_{1}({TX})=0\in H^{2}(X,{{\bf R}})$, and any Kähler class $[\omega]\in H^{2}(X,{{\bf R}})$ there exists a unique Calabi-Yau metric $g_{CY}$ with the class $[\omega]$⁴⁴ 4 Notice that there is a discrepancy in terminology. In algebraic situation one usually calls Calabi-Yau a projective variety with the trivial canonical class in $Pic(X)$, and the polarization is not considered as a part of data.. Up to now, there is no explicitly known non-flat Calabi-Yau metric on a compact manifold.

In Mirror Symmetry one studies the limiting behavior of $g_{CY}$ as the complex structure on $X$ approaches a “cusp” in the moduli space of complex structures (“maximal degeneration”). Well-known conjecture of Strominger, Yau and Zaslow (see [SYZ]) claims a torus fibration structure of Calabi-Yau manifolds near the cusp. A metric approach to the maximal degeneration (see [GW], [KoSo]) explains the structure of such Calabi-Yau manifolds in terms of their Gromov-Hausdorff limits. We recall this picture below following [KoSo].

We start with the definition of a maximally degenerating family of algebraic Calabi-Yau manifolds.

Let ${{\bf C}}_{t}^{mer}=\{f=\sum_{n\geq n_{0}}a_{n}t^{n}\}$ be the field of germs at $t=0$ of meromorphic functions in one complex variable, and ${X}_{mer}$ be an algebraic $n$-dimensional Calabi-Yau manifold over ${{\bf C}}_{t}^{mer}$ (i.e. ${X}_{mer}$ is a smooth projective manifold over ${{\bf C}}_{t}^{mer}$ with the trivial canonical class: $K_{{X}_{mer}}=0$). We fix an algebraic non-vanishing volume element $\Omega\in\Gamma({X}_{mer},K_{{X}_{mer}})$. The pair $({X}_{mer},\Omega)$ defines a 1-parameter analytic family of complex Calabi-Yau manifolds $(X_{t},\Omega_{t}),0<|t|<\epsilon$, for some $\epsilon>0$.

Let $[\omega]\in H^{2}_{DR}({X}_{mer})$ be a cohomology class in the ample cone. Then for every $t$, such that $0<|t|<\epsilon$ it defines a Kähler class $\omega_{t}$ on $X_{t}$. We denote by $g_{X_{t}}$ the unique Calabi-Yau metric on $X_{t}$ with the Kähler class $[\omega_{t}]$.

It follows from the resolution of singularities, that as $t\to 0$ one has

for some $C\in{{\bf C}}^{\times},k\in{{\bf Z}},0\leq m\leq n=\dim\,({X}_{mer})$.

Let us rescale the Calabi-Yau metric: $g_{X_{t}}^{new}=g_{X_{t}}/diam(X_{t},g_{X_{t}})^{1/2}$. In this way we obtain a family of Riemannian manifolds $X_{t}^{new}=(X_{t},g_{X_{t}}^{new})$ of diameter $1$.

There is a more precise conjecture (see [KoSo] for the details) which says that outside of $B^{sing}$ the space $X_{t}^{new}$ is metrically close to a torus fibration with flat Lagrangian fibers (integrable system). This torus fibration can be canonically reconstructed (up to a locally constant twist) from the limiting data a)-d).

Conjecture 1 holds for abelian varieties (since $B=B^{sm}$ is a flat torus in this case). It is non-trivial for K3 surfaces (see [GW] for the proof). In 3-dimensional case there is now a substantial progress (see [LYZ]).

The following easy Proposition is well-known.

We will call integral a Monge-Ampère manifold with ${\bf Z}$-affine structure.

In Mirror Symmetry one often has a so-called dual family of Calabi-Yau manifolds associated with the given one. There is no general definition of the dual family, but there are many examples. The following Conjecture (see [KoSo]) formalizes Strominger-Yau-Zaslow picture of Mirror Symmetry:

One can say that Monge-Ampère manifolds with integral affine structures are real analogs of Calabi-Yau manifolds. Conversely, having an integral Monge-Ampère manifold $(Y,g_{Y},\nabla_{Y},(TY)^{{\bf Z}})$ one can construct a torus fibration $TY/(TY)^{{\bf Z}}\to Y$. It is easy to see that the total space of this fibration is in fact a Calabi-Yau manifold (typically non-compact as $Y$ is non-compact too). Rescaling the covariant lattice we can make fibers small (of the size $O((\log|t|)^{-1})$). As we already mentioned, the extended version of Conjecture 1 says that this torus fibration is close (after a locally constant twist) to $X_{t}^{new}$ outside of a “singular” subset.