The Strominger–Yau–Zaslow Conjecture [22], or SYZ Conjecture, explains Mirror Symmetry between Calabi–Yau 3-folds $X,\hat{X}$ in terms of the existence of special Lagrangian fibrations $f:X\rightarrow B$ and $\hat{f}:\hat{X}\rightarrow B$ over the same base $B$, such that for generic $b\in B$ the fibres $f^{-1}(b)$ and $\hat{f}^{-1}(b)$ are dual 3-tori in $X$ and $\hat{X}$ respectively.

The original SYZ Conjecture was motivated by physical reasoning, and was somewhat vague on mathematical details, so that it is not yet clear exactly what the final form of the conjecture ought to be. But it has been obvious from the beginning that many of the problems in formulating and proving the conjecture will have to do with the singular fibres of the fibrations.

Much work has already been done on the SYZ Conjecture by Zharkov [23], Gross [5, 6, 7], Ruan [18, 19, 20, 21] and others. By and large these papers have focussed on difficult questions of global symplectic topology, and have dealt mostly with Lagrangian fibrations, without making great use of the special Lagrangian condition.

The goal of this paper is to build up a (partly conjectural) picture of the singularities of special Lagrangian fibrations $f:X\rightarrow B$ of a Calabi–Yau 3-fold $X$, particularly in the case when $X$ is generic in a suitable sense. We take a different point of view from the authors above, in that we adopt a local geometric approach rather than a global topological one, and we make essential use of the special Lagrangian condition throughout.

One of our main contentions is that generically the singularities of a special Lagrangian 3-fold $N$ in $X$ will be finitely many isolated points $p_{1},\ldots,p_{k}$, and that the best way to understand them is in terms of local models for the singularities in $\mathbin{\mathbb{C}}^{3}$. That is, in a small neighbourhood of each $p_{j}$ in $X$, $N$ will look like a special Lagrangian 3-fold in $\mathbin{\mathbb{C}}^{3}\cong T_{\smash{p_{j}}}X$ with a singularity at 0.

So to study singularities of special Lagrangian 3-folds in Calabi–Yau 3-folds, we should begin by finding and classifying examples of special Lagrangian 3-folds in $\mathbin{\mathbb{C}}^{3}$. The author has made a start on this process in three papers [11, 12, 13] constructing explicit examples of special Lagrangian $m$-folds in $\mathbin{\mathbb{C}}^{m}$, and expects to write others. While these papers are not directly relevant to the present work, the general understanding of special Lagrangian singularities acquired in these papers was very helpful to the author in thinking about special Lagrangian fibrations.

Most other authors writing about the SYZ Conjecture have considered special Lagrangian fibrations $f:X\rightarrow B$ of a Calabi–Yau 3-fold in which the base space $B$ is a smooth 3-manifold, and $f$ is a smooth map. This implies that the discriminant $\Delta_{f}$ (the set of singular fibres) of $f$ is of codimension two in $B$, and that generic singular fibres are singular along a real curve.

My picture is rather different from this. I believe that in the generic case special Lagrangian fibrations $f:X\rightarrow B$ are only piecewise smooth, being continuous but not differentiable on real hypersurfaces in $X$. Furthermore, the discriminant $\Delta_{f}$ is of codimension one in $B$, and all singular fibres have only finitely many singular points.

Whilst I do not prove this conclusively — this would require some rather difficult analytic results which I hope to prove one day — I do present a lot of evidence in favour of my ideas. As the view that special Lagrangian fibrations are smooth has appeared in so many papers, I also felt it necessary to explain why I think this is wrong. I hope the authors I have disagreed with in person will take this as a mark of respect, as I have learnt a lot from their papers.

We begin in §2 with an introduction to special Lagrangian geometry. We include a discussion of almost Calabi–Yau manifolds and genericity which will probably be unfamiliar to most readers. Section 3 discusses the SYZ Conjecture, explains why generic special Lagrangian fibrations cannot be smooth, and reviews a construction of piecewise smooth Lagrangian fibrations by Wei-Dong Ruan which has many features in common with the picture of special Lagrangian fibrations we will propose.

Section 4 describes two symmetric special Lagrangian fibrations of $\mathbin{\mathbb{C}}^{3}$ due to Harvey and Lawson, and studies their singular fibres. Then §5 defines our local model for codimension one singularities in generic special Lagrangian fibrations. It is an entirely explicit, piecewise smooth special Lagrangian fibration $f:\mathbin{\mathbb{C}}^{3}\rightarrow\mathbin{\mathbb{R}}^{3}$, given by a simple formula. The fibres of $f$ are translations of parts of fibres from the Harvey–Lawson fibration in §4.

We move on in §6–§8 to consider codimension two singularities in special Lagrangian fibrations. We do this by studying a class of $\mathbin{\rm U}(1)$-invariant 3-folds $N$ in $\mathbin{\mathbb{C}}^{3}$ defined in terms of functions $u,v:\mathbin{\mathbb{R}}^{2}\rightarrow\mathbin{\mathbb{R}}$. The condition for $N$ to be special Lagrangian is a p.d.e. on $u$ and $v$, a nonlinear version of the Cauchy–Riemann equations.

Making some conjectures about solutions of these equations, in §7 we describe the expected structure of codimension two singularities. We apply this in §8 to a situation with some nontrivial global topology, and do a monodromy calculation.

Finally, in §9 we review the global properties of smooth special Lagrangian fibrations $f:X\rightarrow B$ uncovered by Mark Gross and Wei-Dong Ruan, and propose how their picture should be modified under a generic perturbation of $X$. Based on this we draw some conclusions, which contradict the stronger current forms of the SYZ Conjecture.