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.