Joyce [14] gave reasons that the above topological picture of Gross-Ruan cannot literally describe a special Lagrangian fibration in a generic Calabi-Yau 3-fold, based on his study of local $U(1)$-invariant special Lagrangian submanifolds inside $\mathbb{C}^{3}$. Joyce’s critique hinges on two geometric observations:

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  Special Lagrangian fibrations need not be defined by a smooth map, and the discriminant locus needs not have codimension 2.

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  The $I_{1}\times S^{1}$ singular fibres have non-isolated special Lagrangian singularities, which is an infinite codimensional phenomenon in the parameter space, namely the singularity structure cannot persist under almost any perturbation of the Kähler structure or the boundary data of the special Lagrangian.

Furthermore, in the $U(1)$-invariant setting, Joyce constructed examples illustrating the possibility that fibres with $I_{1}\times S^{1}$ singularities can break up into fibres with a pair of special Lagrangian $T^{2}$ cones. Such fibres lie over a thickened version of the original edges in $\mathfrak{D}$, and in particular the discriminant locus of the SYZ fibration now has codimension 1.

As suggested by Morrison [22] this thickening picture is linked to the description of the negative vertex (Section 1.1.4) as follows. We can view $B\times T^{2}$ as $(\mathbb{C}^{*})_{z_{1},z_{2}}^{2}\times\mathbb{R}\simeq\mathbb{R}^{2}\times\mathbb{R}\times T^{2}$. The ‘pair of pants’ $S$ is realised topologically by

$$\{z_{1}+z_{2}=1\}\subset(\mathbb{C}^{*})_{z_{1},z_{2}}^{2}=(\mathbb{C}^{*})^{2}\times\{0\}\subset(\mathbb{C}^{*})^{2}\times\mathbb{R}.$$

The algebraic 2-torus $(\mathbb{C}^{*})^{2}$ maps to $\mathbb{R}^{2}_{y_{1},y_{2}}$ via

$$y_{1}=-\frac{1}{2\pi}\log|z_{1}|,\quad y_{2}=-\frac{1}{2\pi}\log|z_{2}|.$$

The image of $S$ in $\mathbb{R}^{2}_{y_{1},y_{2}}$ under this map is an amoeba which can be thought as a thickend version of $\mathfrak{D}\subset\mathbb{R}^{2}_{y_{1},y_{2}}$. Along the 3 directions defined by $\mathfrak{D}$, the asymptotic geometry of $S$ near infinity approaches 3 cylinders. In the modified construction $M^{-}$ is a singular $S^{1}$-bundle over $(\mathbb{C}^{*})^{2}\times\mathbb{R}$ whose fibres collapse to points along the codimension 3 locus $S\subset(\mathbb{C}^{*})^{2}\times\mathbb{R}$. The natural smooth map $M^{-}\to(\mathbb{C}^{*})^{2}\times\mathbb{R}\to B=\mathbb{R}^{2}_{y_{1},y_{2}}\times\mathbb{R}$ cannot be exactly a special Lagrangian fibration since its discriminant locus is of codimension 1, but it is still possible to be an approximate special Lagrangian fibration.

We also wish to resolve a paradox here in advance. Part of our plan is to construct a family of $T^{2}$-symmetric Ooguri-Vafa type Calabi-Yau metrics on the positive vertex, which admit a special Lagrangian fibration with all the topological features predicted by Gross and Ruan, and in particular the singular fibres will have non-isolated singularities. We emphasize there is no contradiction with Joyce’s critique: it is possible for the Ooguri-Vafa type metrics to be a good metric model for a generic Calabi-Yau 3-fold near the large complex structure limit, while the singularity structure of the SYZ fibration changes drastically. Joyce’s critique does not rule out the Gross-Ruan picture as a limiting description of SYZ fibrations.