Right after the introduction of the SYZ conjecture in 1996, a great deal of effort was input into constructing special Lagrangian torus fibrations, or SYZ fibrations, on Calabi-Yau manifolds.²² 2 In [66, 67], instead of constructing such fibrations, Gross assumed their existence and deduced interesting consequences which were predicted by mirror symmetry. Zharkov [173] was the first to construct topological torus fibrations on Calabi-Yau hypersurfaces in a smooth projective toric variety $\mathbb{P}_{\Delta}$. This includes the important example of the quintic 3-fold. Zharkov obtained his fibrations by deforming the restriction of the moment map on $\mathbb{P}_{\Delta}$ to the boundary $\partial\Delta$ of the moment polytope to a nearby smooth Calabi-Yau hypersurface.

Using similar ideas and introducing a gradient-Hamiltonian flow, W.-D. Ruan constructed Lagrangian torus fibrations on quintic 3-folds in a series of papers [141, 142, 143]. He also carried out an important computation of the monodromy of the fibrations, which was later used by Gross [69] to work out a topological version of SYZ mirror symmetry for Calabi-Yau manifolds including the quintic 3-fold. There is also a related work of Mikhalkin [127] which produces smooth torus fibrations on hypersurfaces in toric varieties by applying tools from tropical geometry.

In general, the construction of special Lagrangian fibrations, or even special Lagrangian submanifolds, is a very difficult problem. One promising approach in constructing special Lagrangians is using the mean curvature flow. Thomas [155] formulated a notion of stability for classes of Lagrangian submanifolds with Maslov index zero in a Calabi-Yau manifold, which should be mirror to the stability of holomorphic vector bundles. In [156], Thomas and Yau conjectured that there should exist a unique special Lagrangian representative in a Hamiltonian isotopy class if and only if the class is stable, and that such a representative could be obtained by the mean curvature flow where long time existence should hold. They further proposed a Jordan-Hölder–type decomposition for special Lagrangian submanifolds and related this to formation of singularities in the mean curvature flow. Their proposals and conjectures have a big influence on the development of Calabi-Yau geometry and the SYZ conjecture. There has been a lot of advances in this area [28, 29, 95, 108, 130, 131, 133, 147, 148, 149, 161, 162, 163, 164]; see the excellent survey articles [165] and [132] and references therein for more details. Unfortunately, at the time of writing, it is still unknown whether there exists a special Lagrangian torus fibration on the quintic 3-fold.

Noncompact examples of special Lagrangian fibrations are much easier to come by. Harvey and Lawson’s famous paper on calibrated geometries [85] gave the simplest of such examples: the map defined by

	$\displaystyle f:\mathbb{C}^{3}$	$\displaystyle\to\mathbb{R}^{3},$
	$\displaystyle(z_{1},z_{2},z_{3})$	$\displaystyle\mapsto\left(\textrm{Im}(z_{1}z_{2}z_{3}),|z_{1}|^{2}-|z_{2}|^{2},|z_{1}|^{2}-|z_{3}|^{2}\right),$

is a special Lagrangian fibrations whose fibers are invariant under the diagonal $T^{2}$-action on $\mathbb{C}^{3}$. This example was later largely generalized by independent works of Goldstein [61] and Gross [68]. They constructed explicit special Lagrangian torus fibrations on any toric Calabi-Yau $n$-fold (which are necessarily noncompact) via the compatible $T^{n-1}$-action which preserves the natural holomorphic volume form. The discriminant loci of these examples are of real codimension two and can be described explicitly.

Another set of noncompact examples, which has a historic impact on the development of SYZ mirror symmetry and special Lagrangian geometry, was discovered by Joyce [94]. It was once believed that special Lagrangian fibrations would always be smooth and hence have codimension two discriminant loci. But the examples of Joyce showed that this is unlikely the case. He constructed explicit $S^{1}$-invariant special Lagrangian fibrations which are only piecewise smooth and have real codimension one discriminant loci. The set of singular points of such a fibration is a Riemann surface and its amoeba-shaped image gives the codimension one discriminant locus. Joyce also argued that his examples exhibited the generic behavior of discriminant loci of special Lagrangian fibrations.

This pioneering work of Joyce significantly deepens our understanding of the singularities of special Lagrangian fibrations, and at the same time forces us to rethink about the formulation of the SYZ conjecture. Originally, we expect that a mirror pair of Calabi-Yau manifolds $X$ and $\check{X}$ should have special Lagrangian torus fibrations to the same base $B$ so that their discriminant loci coincide. But the examples of Joyce demonstrate that while the discriminant locus on one side may be of codimension one, that on the other side can be of codimension two. The best that one can hope for is that both $X$ and $\check{X}$ admit special Lagrangian torus fibrations to the same base $B$ and as one approaches the large complex structure limits on both sides, the discriminant loci of these fibrations converge to the same limit which is of codimension two.

More precisely, let $\mathfrak{X}\to D$ and $\check{\mathfrak{X}}\to D$ be maximally unipotent degenerations of Calabi-Yau manifolds mirror to each other, where $D$ is the unit disk and $0\in D$ corresponds to large complex structure limits for the mirror pair. We choose a sequence $\{t_{i}\}\subset D$ converging to 0, and let $g_{i}$ and $\check{g}_{i}$ be Ricci-flat metrics on $\mathfrak{X}_{t_{i}}$ and $\check{\mathfrak{X}}_{t_{i}}$ respectively normalized so that they have fixed diameters $C$. Then we expect that

• (i)

  there are convergent subsequences of $(\mathfrak{X}_{t_{i}},g_{i})$ and $(\check{\mathfrak{X}}_{t_{i}},\check{g}_{i})$ converging (in the Gromov-Hausdorff sense) to metric spaces $(B_{\infty},d_{\infty})$ and $(\check{B}_{\infty},\check{d}_{\infty})$ respectively;

• (ii)

  the spaces $B_{\infty}$ and $\check{B}_{\infty}$ are affine manifolds with singularities which are both homeomorphic to $S^{n}$;

• (iii)

  outside a real codimension 2 locus $\Gamma\subset B_{\infty}$ (resp. $\check{\Gamma}\subset\check{B}_{\infty}$), $d_{\infty}$ (resp. $\check{d}_{\infty}$) is induced by a Monge-Ampère metric; and

• (iv)

  the Monge-Ampère manifolds $B_{\infty}\setminus\Gamma$ and $\check{B}_{\infty}\setminus\check{\Gamma}$ are Legendre dual to each other.

This limiting version of the SYZ conjecture was proposed independently by Gross-Wilson [83] and Kontsevich-Soibelman [105]. In fact, the general question of understanding the limiting behavior of Ricci-flat metrics was raised by Yau in his famous lists of open problems [170, 171]. Motivated by the SYZ picture of mirror symmetry, this question has been studied extensively in the last 15 years, and substantial progress has been made by Gross-Wilson [83], Tosatti [157, 158], Ruan-Zhang [144], Zhang [172], Rong-Zhang [140, 139] and more recently, Gross-Tosatti-Zhang [82, 81].

The metric spaces $B_{\infty}$ and $\check{B}_{\infty}$ should be thought of as limits of bases of SYZ fibrations on the families of Calabi-Yau manifolds. Applying the above picture, one may try to construct the mirror of a maximally unipotent degeneration of Calabi-Yau manifolds $\mathfrak{X}\to D$ as follows. We first identify the Gromov-Hausdorff limit $B_{\infty}$. Then we take the Legendre dual $\check{B}_{0}$ of $B_{\infty}\setminus\Gamma$ and try to compactify the quotient $\check{X}_{0}:=T\check{B}_{0}/\Lambda$ to get the correct mirror. Unfortunately this naïve approach will not work because the natural complex structure on $\check{X}_{0}$ is not globally defined due to nontrivial monodromy of the affine structure around the singularities in $\check{B}_{\infty}$. To get the corrected mirror, one needs to deform the complex structure on $\check{X}_{0}$ by taking into account contributions from holomorphic disks.