If the speculations of §9.2 are correct, they have important consequences for the SYZ Conjecture. Positive and negative singular fibres are expected to be dual to one another under the mirror transform. That is, if we have dual smooth SL fibrations $f:X\rightarrow B$ and $\hat{f}:\hat{X}\rightarrow B$ as in the SYZ conjecture, then positive vertices in the discriminant $\Delta_{f}$ of $f$ in $B$ should coincide with negative vertices in the discriminant $\Delta_{\smash{\hat{f}}}$ of $\hat{f}$, and vice versa. One way to see this is that the monodromy matrices in (69) are the transposes of those in (70).

However, after a small generic perturbation of $f$ and $\hat{f}$ near such a vertex in $B$, it is clear from Figures 4 and 5 that the discriminant loci $\Delta_{f}$ and $\Delta_{\smash{\hat{f}}}$ can no longer be identified, because they are not homeomorphic. On this basis we make the following conjecture.

This is a kind of counter-conjecture to the SYZ Conjecture, in that it contradicts some of the stronger forms of the SYZ Conjecture that people have written down so far, and if it is true then it will limit the scope of any eventual final formulation of the SYZ Conjecture. The author’s feeling is that while the SYZ Conjecture is clearly morally and spiritually true, it is probably not literally true of genuine special Lagrangian fibrations of holonomy $\mathop{\rm SU}(3)$ Calabi–Yau 3-folds, except in some limiting sense in the large complex structure limit.

Finally, we note that the discussion above is based on optimistic assumptions on how well-behaved generic special Lagrangian fibrations are. Here are two ways in which things might go wrong.

• (i)

  Rather than speaking of a fibration $f$, we should instead consider a 3-dimensional family $\mathcal{M}$ of special Lagrangian 3-folds $N$ in $X$, generically 3-tori, and thought of as the fibres of $f$. Hopefully $\mathcal{M}$ is homeomorphic to a compact 3-manifold without boundary.

  It might be that in some regions of $X$ there is more than one SL 3-fold in $\mathcal{M}$ passing through each point. In this case, there will be no map $f:X\rightarrow B$ with fibres $\mathcal{M}$. But $\mathcal{M}$ could still have the property that for each generic point $x$ in $X$ the number of elements of $\mathcal{M}$ passing through $x$, counted with signs, is one, so that $\mathcal{M}$ could be regarded as a ‘fibration’ in a generalized sense.

• (ii)

  Again, we think of the family $\mathcal{M}$ rather than the fibration $f$. But something worse than (i) might happen. Perhaps there is some new kind of codimension one singularity which means that $\mathcal{M}$ is a manifold with boundary. The singularities of §7–§8 do not count as boundary singularities, as $\mathcal{M}$ extends on both sides of them.

  If $\mathcal{M}$ is a manifold with boundary then the number of elements of $\mathcal{M}$ passing through $x\in X$, even counted with signs, need not be constant, and some points $x$ might not lie in any $N\in{\mathcal{M}}$ at all. So the fibration $f$ would not exist even in the generalized sense above.

The author expects (i) to actually be the rule rather than the exception in general Calabi–Yau 3-folds, but perhaps it does not happen close to the large complex structure limit. Behaviour as in part (i) can arise in a nonsingular part of the ‘fibration’, when the harmonic 1-forms on a nonsingular special Lagrangian $T^{3}$ develop zeros, so that neighbouring nonsingular ‘fibres’ intersect one another.

It may also be that the suggestions in part (b) of §9.2 are wrong, and behaviour as in part (ii) above happens instead. It is because the author takes this possibility seriously that we have not made any conjectures that positively assert the existence of special Lagrangian fibrations on generic almost Calabi–Yau 3-folds in this paper, even though it was very tempting to do so.