In this section we propose a mathematical language for the geometric mirror symmetry, understood as a duality of torus fibrations.

Monge-Ampère manifolds were studied (under a different name) in [CY] where is was proven that if $Y$ is compact then its finite cover is a torus.

Let us consider a (non-compact) example motivated by the mirror symmetry for K3 surfaces (see also [GW]). Let $S$ be a complex surface endowed with a holomorphic non-vanishing volume form $vol_{S}$, and $\pi:S\to C$ be a holomorphic fibration over a complex curve $C$, such that fibers of $\pi$ are non-singular elliptic curves.

We define a metric $g_{C}$ on $C$ as the Kähler metric associated with the $(1,1)$-form $\pi_{\ast}(vol_{S}\wedge\overline{vol}_{S})$. Let us choose (locally on $C$) a basis $(\gamma_{1},\gamma_{2})$ in $H_{1}(\pi^{-1}(x),{{\bf Z}}),x\in C$. We define two closed 1-forms on $C$ by the formulas

It follows that $\alpha_{i}=dx_{i}$ for some functions $x_{i},i=1,2$. We define an affine structure on $C$, and the corresponding connection $\nabla$, by saying that $(x_{1},x_{2})$ are affine coordinates. One can check directly that $(C,g_{C},\nabla)$ is a Monge-Ampère manifold. In a typical example of elliptic fibration of a K3 surface, one gets $C={{\bf C}}{\bf P}^{1}\setminus\{z_{1},...,z_{24}\}$, where $\{z_{1},...,z_{24}\}$ is a set of distinct $24$ points in ${{\bf C}}{\bf P}^{1}$.

Returning to the general case, we can restate a portion of our conjectures by saying that the smooth part of the Gromov-Hausdorff limit of a maximally degenerate family of Calabi-Yau manifolds is a Monge-Ampère manifold with an integral affine structure.

There is a well-known duality on local solutions of the Monge-Ampère equation.

Proof. The graph of $L=dK$ is a Lagrangian submanifold in $T^{\ast}{{\bf R}}^{n}={{\bf R}}^{n}\oplus({{\bf R}}^{n})^{\ast}$. Let $p_{1}$ and $p_{2}$ be the natural projections to the direct summands. They are local diffeomorphisms. Since $K$ is defined up to the adding of an affine function, the graph itself is defined up to translations. The Monge-Ampère equation corresponds to the condition $p_{1}^{\ast}(vol_{{\bf R}^{n}})=p_{2}^{\ast}(vol_{{\bf R}^{\ast n}})$, where $vol_{{\bf R}^{n}}$ (resp. $vol_{{\bf R}^{\ast n}}$) denotes the standard volume form on ${\bf R}^{n}$ (resp. ${\bf R}^{\ast n}$). The manifold $L$ can be considered as a graph of $d\widehat{K}$. Thus $\widehat{K}$ satisfies the Monge-Ampère equation as well. The Lemma is proved. $\blacksquare$

The manifold $L$ carries a Riemannian metric $g_{L}$ induced by the indefinite metric $\sum_{i}dx_{i}\,dy_{i}$ on ${{\bf R}}^{n}\oplus({{\bf R}}^{n})^{\ast}$. This metric is given by the matrix $(\partial^{2}K/\partial x_{i}\partial x_{j})$ in coordinates $(x_{1},...,x_{n})$, and by the matrix $(\partial^{2}\widehat{K}/\partial y_{i}\partial y_{j})$ in the dual coordinates. Thus on $L$ we have a metric, and two affine structures (pullbacks of the standard affine structures on the coordinate spaces). Hence we have two structures of the Monge-Ampère manifold on $L$. It is easy to see that the local pictures can be glued together. This leads to the following result.

Now we can state the geometric counterpart of the mirror symmetry conjecture.

Monge-Ampère manifolds with integral affine structures are real analogs of Calabi-Yau manifolds. In fact the mirror duality in the sense of this section holds for a larger class of manifolds. We define an AK-manifold (AK stands for affine and Kähler) as in the Definition 2, but dropping the condition c) (Monge-Ampère equation), see also [CY]. The reader can check easily that all constructions of this section, including the duality of torus fibrations hold for AK-manifolds as well.