Let ${{\bf C}}_{q}^{mer}=\{f=\sum_{n\geq n_{0}}a_{n}q^{n}\}$ be the field of germs at $q=0$ of meromorphic functions in one variable.

Let ${\cal X}_{mer}$ be an algebraic $n$-dimensional Calabi-Yau manifold over ${{\bf C}}_{q}^{mer}$ (i.e. ${\cal X}_{mer}$ is a smooth projective manifold over ${{\bf C}}_{q}^{mer}$ with the trivial canonical class: $K_{{\cal X}}=0$). We fix an algebraic non-vanishing volume element $vol\in\Gamma({\cal X}_{mer},K_{{\cal X}})$. The pair $({\cal X}_{mer},vol)$ defines a 1-parameter analytic family of complex Calabi-Yau manifolds $(X_{q},vol_{q}),0<|q|<r_{0}$, for some $r_{0}>0$.

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

It follows from the resolution of singularities, that as $q\to 0$ one has the following formula:

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

One can show easily that this definition is equivalent to the usual one, given in terms of variations of Hodge structures (see [Mo], [LTY]).

In fact, the vector $v$ in the lemma can be chosen to be proportional to the cohomology class of $vol_{q}$ for any given $q$. Notice that in [De] a slightly stronger condition was imposed: the weight filtration on $H^{*}(X_{q},{\bf C})$ associated with the monodromy operator should be complementary to the Hodge filtration.

Let us recall the definition of the Gromov-Hausdorff metric $\rho_{GH}$. It is a metric on the space of isometry classes of metric spaces of finite diameter. We say that two metric spaces $M_{1}$ and $M_{2}$ are $\varepsilon$-close in $\rho_{GH}$ if there exists a metric space $M$ containing both $M_{1}$ and $M_{2}$ as metric subspaces, such that $M_{1}$ belongs to the $\varepsilon$-neighborhood of $M_{2}$ and vice versa.

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

At the end of this section we propose a non-rigorous explanation of our conjecture based on differential-geometric considerations.

The conjecture opens the way for compactification of the moduli space of Calabi-Yau metrics on a given Calabi-Yau manifold $M$, by adding as a boundary component the set of pairs $(\overline{Y},{{\bf R}}_{+}^{\ast}\cdot g_{\overline{Y}})$ for all 1-parameter maximal degenerations $X_{q}$, such that $X_{q^{\prime}}=M$ for some $q^{\prime}$. This corresponds to a choice of a “cusp” in the moduli space of Calabi-Yau manifolds. This choice is usually described in terms of certain algebro-geometric data: the action of the monodromy operator, variation of Hodge structures, mixed Hodge structure of the special fiber, etc. The previous conjecture offers a pure “metric” description of a cusp.

It follows from part b) of the conjecture that one can choose a $\nabla$-covariant lattice $T_{Y,y}^{{\bf Z}}\subset T_{Y,y},y\in Y$. Suppose we are given a triple $(Y,g_{Y},\nabla)$, satisfying the properties a)-c) of the conjecture, and we have fixed a covariant lattice $T_{Y}^{{\bf Z}}$ in the tangent bundle $T_{Y}$. Then we can construct a 1-parameter family of non-compact complex Calabi-Yau manifolds, endowed with Ricci flat Kähler metrics. Namely, let $X^{\varepsilon}$ be the total space of the torus bundle $p_{\varepsilon}:X^{\varepsilon}\to Y$ with fibers $T_{Y,y}/\varepsilon T_{Y,y}^{{\bf Z}},y\in Y,0<\varepsilon\leq\varepsilon_{0}$. The total space $TY$ of the tangent bundle $T_{Y}$ carries a canonical complex structure coming from the isomorphism $T_{TY}\simeq\pi^{\ast}T_{Y}\oplus\pi^{\ast}T_{Y}\simeq\pi^{\ast}T_{Y}\otimes{{\bf C}}$ where $\pi:TY\longrightarrow Y$ is the canonical projection (here we use the affine structure on $Y$). Using the same identification, we introduce a metric on $TY$, namely $g_{TY}=\pi^{\ast}g_{Y}\oplus\pi^{\ast}g_{Y}$. It is easy to see, that $g_{TY}$ is a Kähler metric with the potential $\pi^{\ast}K$. It follows from the Monge-Ampère equation that the metric $g_{TY}$ is Ricci flat. Passing to the quotient, we obtain on $X^{\varepsilon}$ a complex structure $J_{X^{\varepsilon}}$ and a Ricci flat Kähler metric $g_{X^{\varepsilon}}$.

Let $U\subset Y$ be an open simply-connected subset. Then there is an action of the torus $T^{n}\simeq T_{Y,y}/T_{Y,y}^{{\bf Z}}$ on $p_{\varepsilon}^{-1}(U),y\in U$ (different tori are identified for different points $y\in U$ by means of the connection $\nabla$). It implies that for any $t\in H^{1}(Y,(T_{Y}/T_{Y}^{{\bf Z}})^{discr})$ (cohomology with coefficients in the local system of tori considered as abstract groups) one can define a twisted manifold $X^{\varepsilon,t}$, which is the total space of the torus fibration $p_{\varepsilon,t}:X^{\varepsilon,t}\to Y$.

Roughly speaking, the next conjecture says that the “leading asymptotic term” of the family of Calabi-Yau manifolds $X_{q}^{new},q=e^{-1/\varepsilon}$ near the point of maximal degeneration $\varepsilon=0$, is isomorphic up to a twist to the family $(X^{\varepsilon},J_{X^{\varepsilon}})$ associated with the torus bundle described above.

More precisely, we formulate it as follows.

There is the following motivation for the Conjectures 1 and 2. In general, for a degenerating family of Riemannian metrics with non-negative Ricci curvature, one expects a description in terms of a tower of fibrations (collapses) with singularities (compare with 2.3). ²² 2 Some steps in the program of compactification of the space of metrics are accomplished now (see e.g. [CC]), but still there are many non-clarified issues. In the case of Kähler manifolds there are two basic pictures of a simple collapse. The first case is when both the base and the fiber are Kähler manifolds. In the second case fibers are flat totally real tori of dimension $m$ and the base looks locally as a product of a domain in ${{\bf R}}^{m}$ with a Kähler manifold. The logarithmic factor in the asymptotic behavior of the volume should come only from torus fibers. Thus, the largest possible power of the logarithm can appear only when we have a tower of purely torus fibrations. It seems that the fixing (up to a scalar) of the Kähler class forbids the multiple collapse. These considerations give an intuitive “explanation” of our conjectures.

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.

Considerations from CFT and from differential geometry indicate that the integral affine structure on $Y$ does not depend on the choice of the Kähler class of Calabi-Yau metrics. Thus, we obtain a “combinatorial” invariant $(Y,T^{{\bf Z}}_{Y})$ of (maximally degenerating) Calabi-Yau variety over the local field $K={{\bf C}}((q))$. One can argue that in this case there will be a canonical atlas of coordinate charts such that the transition maps belong to the group $SAff(n,{\bf Z}):=SL(n,{\bf Z})\ltimes{\bf Z}^{n}$. The natural question arises whether one can define and calculate it purely algebraically, without the use of transcendental methods and Calabi-Yau metrics. We expect that the answer to this question is positive. In other words there exists a canonical way to associate the data $(Y,T^{{\bf Z}}_{Y})$ with arbitrary smooth projective variety $X,\,c_{1}(T_{X})=0$ having “maximal degeneration” over an arbitrary field $K$ with a discrete valuation.

The conjectural answer (only for the compactification $\overline{Y}$ of $Y$) is the following: let us choose (after an extension of the field $K$) a model with stable reduction. Call an irreducible component $D$ of the special fiber $X_{0}$ essential if the order of pole at $D$ of the global volume element on $X$ is maximal among all components of $X_{0}$. We define topological space $\overline{Y}(X_{0})$ as the Clemens complex spanned by essential divisors (see [LTY]). Roughly speaking, $k$-cells of $\overline{Y}$ correspond to irreducible components of $(k+1)$-fold intersections of essential divisors. Recently one of us (M.K.) proved, using ideas from motivic integration and from Berkovich theory of non-archimedean analytic spaces (see [Be]), that for different choices of models with stable reduction spaces $\overline{Y}(X_{0})$ can be canonically identified . In examples coming from toric geometry the space $\overline{Y}=\overline{Y}(X_{0})$ is always a manifold.

It is not clear yet what is the origin of the smooth part $Y\subset\overline{Y}$, and of the affine structure on it. Conjecturally, all this comes from a map $\pi:X(\overline{K})\to\overline{Y}$ where $\overline{K}$ is the algebraic closure of $K$. In the differential-geometric picture of torus fibrations (when $K={\bf C}_{mer}$) the map $\pi$ is obvious: it associates with a meromorphic (finitely ramified) family of points $x_{q}\in X_{q}$ the limit point $lim_{q\to 0}\,x_{q}\in Y$ in the metric sense. Also, the differential-geometric picture suggests that the closure of the image $\pi(Z(\overline{K}))$ where $Z\subset K$ is an algebraic subvariety, should be a piecewise linear closed subset of $Y$, and linear pieces of it have rational directions. In particular, if $Z$ is a curve then $\pi(Z(\overline{K}))$ is a graph in $Y$. This opens a way to express Gromov-Witten invariants of $X$ in terms of the Feynman expansion for certain quantum field theory on $Y$.

Also, we expect that the choice of an ample class in $NS(X)\otimes{\bf R}$ on $X$ gives rise to the dual integral affine structure on $Y$ defined again in some purely algebro-geometric way. If the ample class is the first Chern class of a line bundle, then there should be also a canonical reduction of the dual integral affine structure to a $SAff(n,{\bf Z})$ -structure.