We shall now define two piecewise smooth special Lagrangian fibrations $f:\mathbin{\mathbb{C}}^{3}\rightarrow\mathbin{\mathbb{R}}^{3}$ with singular fibres of codimension one in $\mathbin{\mathbb{R}}^{3}$. These will be our local models for the most generic kind of singularity in special Lagrangian fibrations of generic Calabi–Yau 3-folds. We begin by defining a family of SL 3-folds $N_{a,c}$ in $\mathbin{\mathbb{C}}^{3}$ depending on $a\in\mathbin{\mathbb{R}}$ and $c\in\mathbin{\mathbb{C}}$.

These 3-folds $N_{a,c}$ are the fibres of a special Lagrangian fibration of $\mathbin{\mathbb{C}}^{3}$.

Using the equations $|z_{1}|^{2}=z_{1}\bar{z}_{1}$ and $|z_{2}|^{2}=z_{2}\bar{z}_{2}$ to rewrite these expressions gives the first case of (25), the third case when $z_{2}\neq 0$, and the fourth case when $z_{1}\neq 0$. If $z_{1}z_{2}=0$ on the other hand, in each of parts (i)–(iii) of Definition 5 we have $|z_{3}-c|^{2}=0$, so $c=z_{3}$, giving the second case of (25), the third case when $z_{2}=0$, and the fourth case when $z_{1}=0$.

So, if $(z_{1},z_{2},z_{3})\in N_{a,c}$ then we can recover $a$ and $c$ from $(z_{1},z_{2},z_{3})$ as in the theorem. Conversely, for any $(z_{1},z_{2},z_{3})$ in $\mathbin{\mathbb{C}}^{3}$, defining $a,c$ by (24)–(25) and reversing the proof above, we find that $(z_{1},z_{2},z_{3})\in N_{a,c}$. Hence $f^{-1}(a,\mathop{\rm Re}c,\mathop{\rm Im}c)=N_{a,c}$, and $f$ is a special Lagrangian fibration of $\mathbin{\mathbb{C}}^{3}$. $\square$

The singular fibres of the fibration are $N_{0,c}$ for $c\in\mathbin{\mathbb{C}}$, which is singular only at $(0,0,c)$. Thus the set of singular points of singular fibres of the fibration is $\bigl\{(0,0,c):c\in\mathbin{\mathbb{C}}\bigr\}$. Note that this is the same as in Corollary 4.2, and is a complex curve in $\mathbin{\mathbb{C}}^{3}$.

However, $f$ is not smooth on the whole real hypersurface $|z_{1}|=|z_{2}|$, which includes the set of singular points but many other points as well. Thus $f$ fails to be smooth not only at singular points of singular fibres, but also at nonsingular points of singular fibres. We should understand the non-smoothness of $f$ as being related not to a singularity at the point in question, but to a change in the global topology of the whole fibre.

Let us consider the symmetries of the fibration. The fibres $N_{a,c}$ were defined as translations by $(0,0,c)$ of the SL 3-folds $L_{0}^{+}$, $L_{1,a}^{+}$ and $L_{2,-a}^{+}$ defined in §4, and we know that these have symmetry group $\mathbin{\rm U}(1)^{2}$ in $\mathop{\rm SU}(3)$. However, because this $\mathbin{\rm U}(1)^{2}$-action doesn’t commute with the $(0,0,c)$ translations, the subgroup of $\mathop{\rm SU}(3)\ltimes\mathbin{\mathbb{C}}^{3}$ preserving every fibre is smaller: it is $\mathbin{\rm U}(1)$, acting by

Note that the moment map of this action is $|z_{1}|^{2}-|z_{2}|^{2}$, which is $a$ in (24). This is as one would expect, because Lagrangian submanifolds must lie in level sets of the moment maps of their symmetry groups by [11, Prop. 4.2].

The subgroup of $\mathop{\rm SU}(3)\ltimes\mathbin{\mathbb{C}}^{3}$ preserving the fibration, but acting nontrivially on the set of fibres, is rather larger. It is generated by $\mathbin{\rm U}(1)^{2}$ acting on $\mathbin{\mathbb{C}}^{3}$ as in (13)–(14), and the translations $(z_{1},z_{2},z_{3})\mapsto(z_{1},z_{2},z_{3}+c)$ for $c\in\mathbin{\mathbb{C}}$. The involution $(z_{1},z_{2},z_{3})\mapsto(z_{2},z_{1},z_{3})$ also preserves the fibration and takes $(a,\mathop{\rm Re}c,\mathop{\rm Im}c)\mapsto(-a,\mathop{\rm Re}c,\mathop{\rm Im}c)$, but it does not lie in $\mathop{\rm SU}(3)\ltimes\mathbin{\mathbb{C}}^{3}$ as it changes the sign of $\Omega$.

Now we defined the fibration $f$ and fibres $N_{a,c}$ above using the SL 3-folds $L_{0}^{+}$, $L_{1,a}^{+}$ and $L_{2,-a}^{+}$ of equations (21)–(23). The choice of $+$ rather than $-$ was arbitrary, and we could equally well have used $L_{0}^{-}$, $L_{1,a}^{-}$ and $L_{2,-a}^{-}$ instead. When we do, we get the following analogues of Definition 5 and Theorem 5.2.

The fibres of the fibrations of Theorems 5.2 and 5.4 are singular if and only if $a=0$, that is, on a hyperplane of real codimension one in the base $\mathbin{\mathbb{R}}^{3}$ of the fibrations. But by Proposition 3.2 (which also applies in the noncompact case), if $f$ were a smooth fibration then the set of singular fibres would have Hausdorff codimension at least two. Therefore the piecewise-smoothness of $f$ is essential, not merely cosmetic.

To see what we might mean by a special Lagrangian fibration whose non-smoothness is merely cosmetic, consider the following fairly trivial example.

For each $(a,b,c)\in\mathbin{\mathbb{R}}^{3}$, define $\Pi_{a,b,c}$ to be the affine special Lagrangian 3-plane $\gamma(c)+(a,b,ic)$. It is not difficult to show that there is a unique, continuous special Lagrangian fibration $f:\mathbin{\mathbb{C}}^{3}\rightarrow\mathbin{\mathbb{R}}^{3}$ with $f^{-1}(a,b,c)=\Pi_{a,b,c}$. However, because $\gamma$ is not smooth, $f$ is not smooth.

What is going on here is that we divide $\mathbin{\mathbb{C}}^{3}$ into the family of parallel real hyperplanes $\mathop{\rm Im}(z_{3})=c$, and fibre each such hyperplane by a 2-dimensional family of parallel special Lagrangian 3-planes. There is an ${\mathcal{S}}^{2}$ family $S$ of different ways of fibring $\mathop{\rm Im}(z_{3})=c$ by such parallel 3-planes. We exclude $L$ because then any $L^{\prime}\in S\setminus\{L\}$ is transverse to $\mathbin{\mathbb{R}}^{2}=\bigl\{(a,b,0):a,b\in\mathbin{\mathbb{R}}\bigr\}$, so we can use $a,b$ to parametrize the family of parallel 3-planes in $\mathop{\rm Im}(z_{3})=c$.

The key idea is that we can treat different hyperplanes $\mathop{\rm Im}(z_{3})=c$ essentially independently. The map $\gamma$ is arbitrary; we could choose it to be smooth, or piecewise smooth, or merely continuous, and we then get a fibration $f:\mathbin{\mathbb{C}}^{3}\rightarrow\mathbin{\mathbb{R}}^{3}$ with the same property. So this example generates many examples of piecewise smooth SL fibrations of $\mathbin{\mathbb{C}}^{3}$. However, though Example 5 does show that non-smooth SL fibrations are possible locally, it tells us almost nothing about the smoothness of fibrations of Calabi–Yau 3-folds $X$ by compact special Lagrangian 3-folds $N$, in particular tori.

This is because we know from Theorem 2.9 that if $N$ is a nonsingular SL $T^{3}$ in $X$, then the family of deformations of $N$ is locally smooth and 3-dimensional. Hence, if these deformations are locally transverse to $N$, then near $N$ they do form a smooth SL fibration. Where this argument breaks down is when the fibres of the fibration develop singularities. Thus, any argument as to whether SL fibrations are smooth must focus on the behaviour of the fibrations near their singularities.

As Example 5 involves no singular fibres, it is irrelevant to the discussion. However, Theorems 5.2 and 5.4 are relevant as they model fibrations with many singular fibres, whose singularities are of a kind that cannot appear in smooth fibrations. They are evidence in favour of our contention that special Lagrangian fibrations of Calabi–Yau 3-folds will not in general be smooth.