The discussion of §3 demonstrates that the SYZ conjecture gives a beautiful description of mirror symmetry at a purely topological level. This, by itself, can often be useful, but fails to get at the original hard differential geometric conjecture and fails to give insight into why mirror symmetry counts curves.

In order for the full-strength version of the SYZ conjecture to hold, the strong version of duality for topological torus fibrations we saw in §3 should continue to hold at the special Lagrangian level. This would mean that a mirror pair $X,\check{X}$ would possess special Lagrangian torus fibrations $f:X\rightarrow B$ and $\check{f}:\check{X}\rightarrow B$ with codimension two discriminant loci, and the discriminant loci of $f$ and $\check{f}$ would coincide. These fibrations would then be dual away from the discriminant locus.

There are examples of special Lagrangian fibrations on non-compact toric varieties $X$ with discriminant locus looking very similar to what we have described in the topological case. In particular, if $X$ is an $n$-dimensional Ricci-flat Kähler manifold with a $T^{n-1}$-action preserving the metric and holomorphic $n$-form, then $X$ will have a very nice special Lagrangian fibration with codimension two discriminant locus. (See [21] and [16]). However, Dominic Joyce (see [47] and other papers cited therein) began studying some three-dimensional $S^{1}$-invariant examples, and discovered quite different behaviour. There is an argument in [19] that if a special Lagrangian fibration is $C^{\infty}$, then the discriminant locus will be (Hausdorff) codimension two. However, Joyce discovered examples which were not differentiable, but only piecewise differentiable, and furthermore, had a codimension one discriminant locus:

This example forces a reevaluation of the strong form of the SYZ conjecture. In further work Joyce found evidence for a more likely picture for general special Lagrangian fibrations in three dimensions. The discriminant locus, instead of being a codimension two graph, will be a codimension one blob. Typically the union of the singular points of singular fibres will be a Riemann surface, and it will map to an amoeba-shaped set in $B$, i.e., the discriminant locus looks like the picture on the right rather than the left in Figure 4, and will be a fattening of the old picture of a codimension two discriminant.

Joyce made some additional arguments to suggest that this fattened discriminant locus must look fundamentally different in a neighbourhood of the two basic types of vertices we saw in §3, with the two types of vertices expected to appear pretty much as depicted in Figure 4. Thus the strong form of duality mentioned above, where we expect the discriminant loci of the special Lagrangian fibrations on a mirror pair to be the same, cannot hold. If this is the case, one needs to replace this strong form of duality with a weaker form.

It seems likely that the best way to rephrase the SYZ conjecture is in a limiting form. Mirror symmetry as we currently understand it has to do with degenerations of Calabi-Yau manifolds. Given a flat family $f:\mathcal{X}\rightarrow D$ over a disk $D$, with the fibre $\mathcal{X}_{0}$ over $0$ singular and all other fibres $n$-dimensional Calabi-Yau manifolds, we say the family is maximally unipotent if the monodromy transformation $T:H^{n}(\mathcal{X}_{t},\mathbb{Q})\rightarrow H^{n}(\mathcal{X}_{t},\mathbb{Q})$ ($t\in D$ non-zero) satisfies $(T-I)^{n+1}=0$ but $(T-I)^{n}\not=0$. It is a standard expectation of mirror symmetry that mirrors should be associated to maximally unipotent degenerations of Calabi-Yau manifolds. In particular, given two different maximally unipotent degenerations in a single complex moduli space for some Calabi-Yau manifold, one might obtain different mirror manifolds. Such degenerations are usually called “large complex structure limits” in the physics literature, although sometimes this phrase is used to impose some additional conditions on the degeneration, see [62].

We recall the definition of Gromov-Hausdorff convergence, a notion of convergence of a sequence of metric spaces.

It follows from results of Gromov (see for example [67], pg. 281, Cor. 1.11) that the space of compact Ricci-flat manifolds with diameter $\leq C$ is precompact with respect to Gromov-Hausdorff distance, i.e., any sequence of such manifolds has a subsequence converging with respect to the Gromov-Hausdorff distance to a metric space. This metric space could be quite bad; this is quite outside the realm of algebraic geometry! Nevertheless, this raises the following natural question. Given a maximally unipotent degeneration of Calabi-Yau manifolds $\mathcal{X}\rightarrow D$, take a sequence $t_{i}\in D$ converging to $0$, and consider a sequence $(\mathcal{X}_{t_{i}},g_{t_{i}})$, where $g_{t_{i}}$ is a choice of Ricci-flat metric chosen so that $Diam(g_{t_{i}})$ remains bounded. What is the Gromov-Hausdorff limit of $(\mathcal{X}_{t_{i}},g_{t_{i}})$, or the limit of some convergent subsequence?

This simple example motivates the first conjecture about maximally unipotent degenerations, conjectured independently by myself and Wilson on the one hand [37] and Kontsevich and Soibelman [52] on the other.

Here the topology of the limit depends on the nature of the non-singular fibres $\mathcal{X}_{t}$; for example, if instead $\mathcal{X}_{t}$ was hyperkähler, then we would expect the limit to be a projective space. Also, even in the case of full $SU(n)$ holonomy, if $\mathcal{X}_{t}$ is not simply connected, we would expect limits such as $\mathbb{Q}$-homology spheres to arise.

Conjecture 5.4 is directly inspired by the SYZ conjecture. Suppose we had special Lagrangian fibrations $f_{i}:\mathcal{X}_{t_{i}}\rightarrow B_{i}$. Then as the maximally unipotent degeneration is approached, one can see that the volume of the fibres of these fibrations goes to zero. This would suggest these fibres collapse, hopefully leaving the base as the limit.

This conjecture was proved by myself and Wilson in 2000 for K3 surfaces in [37]. The proof relied on a number of pleasant facts about K3 surfaces. First, they are hyperkähler manifolds, and a special Lagrangian torus fibration becomes an elliptic fibration after a hyperkähler rotation of the complex structure. Since it is easy to construct elliptic fibrations on K3 surfaces, and indeed such a fibration arises from the data of the maximally unipotent degeneration, it is easy to obtain a special Lagrangian fibration. Once this is done, one needs to carry out a detailed analysis of the behaviour of Ricci-flat metrics in the limit. This is done by creating good approximations to Ricci-flat metric, using the existence of explicit local models for these metrics near singular fibres of special Lagrangian fibrations in complex dimension two.

Most of the techniques used are not available in higher dimension. However, much more recently, weaker collapsing results in the hyperkähler case were obtained in work with V. Tosatti and Y. Zhang in [35], assuming the existence of abelian variety fibrations analogous to the elliptic fibrations in the K3 case. Rather than getting an explicit approximate Ricci-flat metric, we make use of a priori estimates of Tosatti in [75].

In the general Calabi-Yau case, the only progress towards the conjecture has been work of Zhang in [77] showing existence of special Lagrangian fibrations in regions of Calabi-Yau manifolds with bounded injectivity radius and sectional curvature and deduces local collapsing from the existence of special Lagrangian fibrations.

The motivation for Conjecture 5.4 from SYZ also provides a limiting form of the conjecture. There are any number of problems with trying to prove the existence of special Lagrangian fibrations on Calabi-Yau manifolds. Even the existence of a single special Lagrangian torus near a maximally unipotent degeneration is unknown, but we expect it should be easier to find them as we approach the maximally unipotent point. Furthermore, even if we find a special Lagrangian torus, we know that it moves in an $n$-dimensional family, but we don’t know its deformations fill out the entire manifold. In addition, there is no guarantee that even if it does, we obtain a foliation of the manifold: nearby special Lagrangian submanifolds may intersect. (For an example, see [59].) So instead, we will just look at the moduli space of special Lagrangian tori.

Given a maximally unipotent degeneration of Calabi-Yau manifolds of dimension $n$, it is known that the image of $(T-I)^{n}:H_{n}(\mathcal{X}_{t},\mathbb{Q})\rightarrow H_{n}(\mathcal{X}_{t},\mathbb{Q})$ is a one-dimensional subspace $W_{0}$. Suppose, given $t_{i}\rightarrow 0$, that for $t_{i}$ sufficiently close to zero, there is a special Lagrangian $T^{n}$ which generates $W_{0}$. This is where we expect to find fibres of a special Lagrangian fibration associated to a maximally unipotent degeneration. Let $B_{0,i}$ be the moduli space of deformations of this torus; every point of $B_{0,i}$ corresponds to a smooth special Lagrangian torus in $\mathcal{X}_{t_{i}}$. This manifold then comes equipped with the McLean metric and affine structures defined in §2. One can then compactify $B_{0,i}\subseteq B_{i}$, (probably by taking the closure of $B_{0,i}$ in the space of special Lagrangian currents; the details aren’t important here). This gives a series of metric spaces $(B_{i},d_{i})$ with the metric $d_{i}$ induced by the McLean metric. If the McLean metric is normalized to keep the diameter of $B_{i}$ constant independent of $i$, then we can hope that $(B_{i},d_{i})$ converges to a compact metric space $(B_{\infty},d_{\infty})$. Here then is the limiting form of SYZ:

Essentially what this is saying is that as we approach the maximally unipotent degeneration, we expect to have a special Lagrangian fibration on larger and larger subsets of $\mathcal{X}_{t_{i}}$. Furthermore, in the limit, the codimension one discriminant locus suggested by Joyce converges to a codimension two discriminant locus, and (the not necessarily Monge-Ampère, see [59]) Hessian metrics on $B_{0,i}$ converge to a Monge-Ampère metric.

The main point I want to get at here is that it is likely the SYZ conjecture is only “approximately” correct, and one needs to look at the limit to have a hope of proving anything. On the other hand, the above conjecture seems likely to be accessible by currently understood techniques. I remain hopeful that this conjecture will be proved, though much additional work will be necessary.

How do we do mirror symmetry using this modified version of the SYZ conjecture? Essentially, we would follow these steps:

1. (1)

   We begin with a maximally unipotent degeneration of Calabi-Yau manifolds $\mathcal{X}\rightarrow D$, along with a choice of polarization. This gives us a Kähler class $[\omega_{t}]\in H^{2}(\mathcal{X}_{t},\mathbb{R})$ for each $t\in D\setminus 0$, represented by $\omega_{t}$ the Kähler form of a Ricci-flat metric $g_{t}$.

2. (2)

   Identify the Gromov-Hausdorff limit of a sequence $(\mathcal{X}_{t_{i}},r_{i}g_{t_{i}})$ where $t_{i}\rightarrow 0$ and $r_{i}$ is a scale factor which keeps the diameter of $\mathcal{X}_{t_{i}}$ constant. The limit will be, if the above conjectures work, an affine manifold with singularities $B$ along with a Monge-Ampère metric.

3. (3)

   Perform a Legendre transform to obtain a new affine manifold with singularities $\check{B}$, though with the same metric.

4. (4)

   Try to construct a compactification of $X_{\epsilon}(\check{B}_{0})$ for small $\epsilon>0$ to obtain a complex manifold $X_{\epsilon}(\check{B})$. This will be the mirror manifold.

As we shall see, we do not expect that we will need the full strength of steps (2) and (3) to carry out mirror symmetry; some way of identifying the base $B$ will be sufficient. Nevertheless, (2) is interesting from the point of view of understanding the differential geomtry of Ricci-flat Kähler manifolds.

Step (4), on the other hand, is crucial, and we need to elaborate on this last step a bit more. The problem is that while we expect that it should be possible in general to construct symplectic compactifications of the symplectic manifold $\check{X}(B_{0})$ (and hence get the mirror as a symplectic manifold, see [8] for the three-dimensional case), we don’t expect to be able to compactify $X_{\epsilon}(\check{B}_{0})$ as a complex manifold. Instead, the expectation is that a small deformation of $X_{\epsilon}(\check{B}_{0})$ is necessary before it can be compactified. Furthermore, this small deformation is critically important in mirror symmetry: it is this small deformation which provides the $B$-model instanton corrections.

Because this last item is so important, let’s give it a name:

We will return to this question later in the paper. However, I do not wish to dwell further on the differential-geometric versions of the SYZ conjecture here. Instead I will move on to describing how the above discussion motivated the algebro-geometric program developed by myself and Siebert for understanding mirror symmetry, and then describe recent work and ideas coming out of this program.