Here is a review of the expected properties of smooth special Lagrangian fibrations of Calabi–Yau 3-folds. Our principal sources are Gross [6, §3] and Ruan [19, §7] for the topology of the singular fibres, and Gross [7, §1] and Ruan [20, §9] for the monodromy matrices. A concise statement may be found in the ‘Precise SYZ mirror conjecture’ of Ruan [20, §9].

Let $f:X\rightarrow{\mathcal{S}}^{3}$ be a smooth special Lagrangian fibration, with fibres $N_{b}=f^{-1}(b)$, and generic fibre $T^{3}$. For generic such fibrations, the discriminant $\Delta_{f}$ is thought to be a trivalent graph, made up of smooth edges, and vertices of two kinds, which we shall refer to as positive and negative. The topology and local monodromy for each kind of singular fibre are as follows.

• (a)

  Let $\gamma$ be an edge in $\Delta_{f}$, and $b\in\gamma$. Then $N_{b}$ has the topology of $T^{3}$ with $T^{2}$ collapsed to an ${\mathcal{S}}^{1}$, and may be written $\Sigma\times{\mathcal{S}}^{1}$, where $\Sigma$ is a $T^{2}$ with an ${\mathcal{S}}^{1}$ collapsed to a point, or equivalently an ${\mathcal{S}}^{2}$ with two points identified. These fibres are called type $(2,2)$ by Gross and type $I$ by Ruan. They have Euler characteristic zero.

  The monodromy about each edge $\gamma$ in $\Delta_{f}$, acting on $H_{1}(T^{3};\mathbin{\mathbb{Z}})\cong\mathbin{\mathbb{Z}}^{3}$, is

  $$\begin{pmatrix}1&1&0\\ 0&1&0\\ 0&0&1\end{pmatrix}$$

  (68)

  with respect to a suitable basis of $H_{1}(T^{3};\mathbin{\mathbb{Z}})$.

• (b)

  Let $b$ be a positive vertex in $\Delta_{f}$. Then $N_{b}$ has the topology of $T^{3}$ with $T^{2}$ collapsed to a point. It has Euler characteristic 1. These fibres are called type $(1,2)$ by Gross and type $III$ by Ruan.

  The monodromies around the three edges $\gamma_{1},\gamma_{2},\gamma_{3}$ meeting at $b$ are

  $$\begin{pmatrix}1&0&0\\ 1&1&0\\ 0&0&1\end{pmatrix},\quad\begin{pmatrix}1&0&0\\ 0&1&0\\ -1&0&1\end{pmatrix}\quad\text{and}\quad\begin{pmatrix}1&0&0\\ -1&1&0\\ 1&0&1\end{pmatrix}$$

  (69)

  with respect to a suitable basis of $H_{1}(T^{3};\mathbin{\mathbb{Z}})$.

  The smooth special Lagrangian fibration of Corollary 4.4 is a local model for the fibration $f$ near the singular point of a positive singular fibre.

• (c)

  Let $b$ be a negative vertex in $\Delta_{f}$. Then Ruan [19, §7] gives two different possible topologies for $N_{b}$, which he calls type $II$ and type $\tilde{II}$. His type $\tilde{II}$ topology agrees with Gross’ proposed type (2,1) fibre [6, §3].

  Both fibres are constructed by taking a fibration $\pi:T^{3}\rightarrow T^{2}$ with fibre ${\mathcal{S}}^{1}$, and collapsing the fibres to points over a graph $\Gamma$ in $T^{2}$. In the type $II$ case $\Gamma$ has three edges and two vertices, and in the type $\tilde{II}$ case it has two edges and one vertex. In both cases $N_{b}$ has Euler characteristic $-1$.

  The monodromies around the three edges $\gamma_{1},\gamma_{2},\gamma_{3}$ meeting at $b$ are

  $$\begin{pmatrix}1&1&0\\ 0&1&0\\ 0&0&1\end{pmatrix},\quad\begin{pmatrix}1&0&-1\\ 0&1&0\\ 0&0&1\end{pmatrix}\quad\text{and}\quad\begin{pmatrix}1&-1&1\\ 0&1&0\\ 0&0&1\end{pmatrix},$$

  (70)

  with respect to a suitable basis of $H_{1}(T^{3};\mathbin{\mathbb{Z}})$.

  At present, to the author’s knowledge, there is no known local model for a smooth special Lagrangian fibration (or even a smooth Lagrangian fibration) in the neighbourhood of a codimension three singular point of a negative singular fibre. It may be that no such local model exists. If this is the case then smooth special Lagrangian fibrations may not exist on general Calabi–Yau 3-folds, even with a very nongeneric choice of almost Calabi–Yau metric.

We will refer to the singular fibres over positive and negative vertices as positive and negative singular fibres respectively. Our notation of positive and negative vertices was suggested by David Morrison, and refers to the sign of the Euler characteristic of the singular fibres. Gross’ notation refers to the Betti numbers $(b^{1},b^{2})$ of the singular fibres.

In the author’s view, none of the fibre topologies in parts (a)–(c) above can occur as special Lagrangian submanifolds in generic almost Calabi–Yau 3-folds. The fibres of parts (a) and (c) are singular along real curves, and so should be highly nongeneric by the argument given in §3.1. Here is an argument to show that a positive singular fibre, a $T^{3}$ with $T^{2}$ collapsed to a point, cannot occur as a special Lagrangian 3-fold in a generic almost Calabi–Yau 3-fold.

Let $N$ be a singular SL 3-fold in $X$ with the topology of $T^{3}$ with $T^{2}$ collapsed to a point. The suspension $S(T^{2})$ of $T^{2}$ is defined to be $T^{2}\times[0,1]$ with the two boundary components $T^{2}\times\{0\}$ and $T^{2}\times\{1\}$ collapsed to two points $p_{0}$ and $p_{1}$. We regard $N$ as an immersion of $S(T^{2})$ in which $p_{0}$ and $p_{1}$ have the same image.

The singularity of $N$ is two $T^{2}$-cones meeting at their vertices. According to the author’s theory of SL singularities mentioned in §7.3, generic SL $T^{2}$-cone singularities are modelled on the isomorphic cones $L_{0}^{\pm}$ of (16). So suppose that the singularity of $N$ is locally modelled on two copies of $L_{0}^{\pm}$.

Now consider how $N$ deforms under small generic perturbations of $X$ as an almost Calabi–Yau 3-fold. According to the author’s theory, a singular SL 3-fold with the topology of $S(T^{2})$ and two singular points modelled on $L_{0}^{\pm}$ should be isolated and stable under small deformations. Thus, as an immersed copy of $S(T^{2})$ we expect $N$ to be stable under deformations of $X$. However, there is no reason for the two singular points $p_{0},p_{1}$ of $S(T^{2})$ to coincide when we deform $X$.

The condition for this to happen is of real codimension 6 in the space of all almost Calabi–Yau 3-folds. Therefore, singular SL 3-folds with the topology of $T^{3}$ with $T^{2}$ collapsed to a point should exist only in codimension 6, and not in the generic case. Thus a positive singular fibre is not a feasible model for singular fibres in generic SL fibrations.

Note also that Ruan’s piecewise smooth Lagrangian fibrations defined using gradient flow, described briefly in §3.2, contain positive singular fibres by [19, Th. 2.2]. So the argument above shows that the topological type of Ruan’s fibrations is not quite right to be fibrations of generic almost Calabi–Yau manifolds, although the degree of nongenericity is much less than in the smooth fibration picture.

The author believes that the Gross–Ruan smooth fibration picture is essentially correct at the (degenerate) large complex structure limit, and perhaps also for some nongeneric almost Calabi–Yau structures near the complex limit. It should therefore be a very valuable tool for understanding the topology and symplectic geometry of mirror Calabi–Yau 3-folds, and will presumably give the right answers. For many of the purposes in which one would apply the Gross–Ruan picture, it is irrelevant whether there actually exists such a genuine special Lagrangian fibration or not.

We are now ready to say something about what special Lagrangian fibrations of generic almost Calabi–Yau 3-folds might look like. Suppose we start with a smooth Gross–Ruan fibration $f:X\rightarrow B$, either of a nongeneric almost Calabi–Yau 3-fold or of the degenerate large complex structure limit, and make a small perturbation to a generic almost Calabi–Yau 3-fold. What happens to the fibration?

Near a nonsingular fibre $N_{b}=f^{-1}(b)$ of $f$, the fibration should remain nonsingular, and the local geometry unchanged. The interesting question is what happens to the singular fibres of $f$. The following is the author’s best guess, on the assumption that special Lagrangian fibrations are well-behaved in the generic case. We preface it with some remarks on monodromy and coordinates on the moduli space.

Let $f:X\rightarrow B$ be an SL fibration with generic fibre $T^{3}$. By Theorem 2.9, near a nonsingular fibre $N_{b}$ the moduli space of deformations of $N_{b}$ is isomorphic to $H^{1}(N_{b};\mathbin{\mathbb{R}})\cong\mathbin{\mathbb{R}}^{3}$. But this moduli space is $B$, and so near any point in $B\setminus\Delta_{f}$ we have natural affine coordinates modelled on $H^{1}(T^{3};\mathbin{\mathbb{R}})$.

However, near a singular fibre $N_{b}$ the situation is more complicated because of the monodromy action. Let $N_{b^{\prime}}$ be a nonsingular fibre near $N_{b}$. Let $\Gamma_{b}$ be the set of monodromies of loops in $B\setminus\Delta_{f}$ based at $b^{\prime}$ and staying in a small neighbourhood of $b$. Then $\Gamma_{b}$ is a group acting on $H_{1}(N_{b^{\prime}};\mathbin{\mathbb{Z}})$ and $H^{1}(N_{b^{\prime}};\mathbin{\mathbb{R}})$. Roughly speaking, near $b$ we can regard $B$ as a kind of quotient of $H^{1}(N_{b^{\prime}};\mathbin{\mathbb{R}})$ by $\Gamma_{b}$, so that $B$ is a kind of orbifold, with the topology of a 3-manifold, but not the smooth structure.

In what follows, as long as we make use of only $\Gamma_{b}$-invariant objects, we can think of $B$ as being locally like $\mathbin{\mathbb{R}}^{3}$ and mostly ignore the monodromy action. We shall represent elements of $H_{1}(N_{b^{\prime}};\mathbin{\mathbb{Z}})$ by column vectors, and elements of $H^{1}(N_{b^{\prime}};\mathbin{\mathbb{R}})$ by row vectors, upon which the monodromy matrices of equations (68)–(70) act by left and right multiplication respectively.

Here is how generic fibrations might work near the perturbation of the Gross–Ruan singular fibres described in parts (a)–(c) of §9.1.

• (a)

  The author conjectures that under small deformations, the ‘edges’ in the Gross–Ruan picture will thicken out into thin ‘ribbons’ of the kind described in §8. They are closed subsets of hyperplanes in $B$ defined locally by $[\omega]\cdot[D]=0$, where $[\omega]$ is the relative de Rham cohomology class in $H^{1}(X,N_{b};\mathbin{\mathbb{R}})$ and $[D]$ a relative homology class in $H_{1}(X,N_{b};\mathbin{\mathbb{Z}})$ depending on the edge, which we expect to be represented by one or more holomorphic discs $D$ for some $b\in B$, as we discussed in §7.3.

  The situation described in §8, in which the generic fibre has two singular points, is only the simplest possibility. In general we expect the generic singular fibres to contain an even number of singular points, divided equally into two kinds. In codimension one on the ribbon these singular points can appear or disappear in pairs of different kinds, and the edge of the ribbon is where the last two singular points disappear.

  We can give local models for such fibrations by modifying Assumption 8.1, replacing the function $\cos x$ in parts (iii) and (iv) by a more general smooth function $g(x)$ with period $2\pi$ and nondegenerate stationary points, modifying part (v) to refer to the stationary points of $g$, and dropping part (vi) entirely.

  Observe that the matrices (67) and (68) are conjugate in $\mathop{\rm SL}(3,\mathbin{\mathbb{Z}})$. Thus, the monodromy around a ‘ribbon’ that we calculated in §8.2 is the same as that around an edge in the Gross–Ruan picture.

• (b)

  For positive vertices in the Gross–Ruan picture, the monodromy matrices of (69) all fix the vectors

  $${\bf v}_{1}=\begin{pmatrix}0\\ 1\\ 0\end{pmatrix},\quad{\bf v}_{2}=\begin{pmatrix}0\\ 0\\ -1\end{pmatrix}\quad\text{and}\quad{\bf v}_{3}=\begin{pmatrix}0\\ -1\\ 1\end{pmatrix}$$

  in $H_{1}(N_{b^{\prime}};\mathbin{\mathbb{Z}})$ and the direction $(1\,0\,0)$ in $H^{1}(N_{b^{\prime}};\mathbin{\mathbb{R}})$.

  In a generic perturbation of a Gross–Ruan fibration near a positive vertex, the three edges in $\Delta_{f}$ should thicken out into ‘ribbons’ $R_{1},R_{2},R_{3}$ lying in the three hyperplanes

  	$$\displaystyle H_{1}=\bigl\{(x_{1},0,x_{3}):x_{j}\in\mathbin{\mathbb{R}}\bigr\},\qquad H_{2}=\bigl\{(x_{1},x_{2},0):x_{j}\in\mathbin{\mathbb{R}}\bigr\}$$
  	$$\displaystyle\text{and}\qquad H_{3}=\bigl\{(x_{1},x_{2},x_{3}):x_{j}\in\mathbin{\mathbb{R}},\quad x_{2}=x_{3}\bigr\},$$

  which are the hyperplanes dual to ${\bf v}_{1},{\bf v}_{2},{\bf v}_{3}$, and intersect in the line $\bigl\{(x_{1},0,0):x_{1}\in\mathbin{\mathbb{R}}\bigr\}$. The ribbons $R_{1},R_{2},R_{3}$ intersect in a bounded subinterval of this line, as sketched in Figure 4.

  There are two obvious ways for this to happen, in which either $R_{1}\cap R_{2}\cap R_{3}$ is part of the boundary of each $R_{j}$, or the ribbons $R_{j}$ extend a little way beyond their intersection $R_{1}\cap R_{2}\cap R_{3}$. The author thinks that the latter option is what actually happens, as in Figure 4.

  For generic points in the intersection $R_{1}\cap R_{2}\cap R_{3}$ the singularities of the fibres are just finitely many points modelled locally on the $T^{2}$-cones $L_{0}^{\pm}$ of (16). These are divided into three kinds, corresponding to the ribbons $R_{1},R_{2},R_{3}$, according to the homology class of the ${\mathcal{S}}^{1}$ in $T^{3}$ that collapses to a point.

  However, at certain special points $b_{0}$ in $R_{1}\cap R_{2}\cap R_{3}$ there will be a new kind of codimension three singularity, when two or three of these singular points of different kinds come together. The author does not have a local model for this singularity, but topologically it may involve a cone on a genus 2 surface. There must be at least one such singular fibre, as it is necessary for the monodromy to work out.

  We expect that when $N_{b^{\prime}}$ is a nonsingular fibre near the ribbon $R_{j}$, there should exist holomorphic discs $D$ in $X$ whose boundary $\partial D$ in $N_{b^{\prime}}$ has homology class ${\bf v}_{j}$ in $H_{1}(N_{b^{\prime}};\mathbin{\mathbb{Z}})$. Singularities develop when the area of $D$ shrinks to zero, which happens on the hyperplane $H_{j}$ in $B$.

• (c)

  For negative vertices, the monodromy matrices of (70) all fix the vector $(1\,0\,0)^{T}$ in $H_{1}(N_{b^{\prime}};\mathbin{\mathbb{Z}})$ and the hyperplane $\bigl\{(0,x_{2},x_{3}):x_{j}\in\mathbin{\mathbb{R}}\bigr\}$ in $H^{1}(N_{b^{\prime}};\mathbin{\mathbb{R}})$. In a generic perturbation of a Gross–Ruan fibration near a negative vertex, the three edges in $\Delta_{f}$ should thicken out into ‘ribbons’ which all lie in the same hyperplane $H$ in $B$, isomorphic to $\bigl\{(0,x_{2},x_{3}):x_{j}\in\mathbin{\mathbb{R}}\bigr\}$ in $H^{1}(N_{b^{\prime}};\mathbin{\mathbb{R}})$. The three ribbons merge together to make a letter $Y$ shape in $H$, as sketched in Figure 5.

  We expect that when $N_{b^{\prime}}$ is a nonsingular fibre near this part of $\Delta_{f}$, there should exist an even number of homologous holomorphic discs $D$ in $X$ whose boundaries $\partial D$ in $N_{b^{\prime}}$ have homology class $(1\,0\,0)^{T}$ in $H_{1}(N_{b^{\prime}};\mathbin{\mathbb{Z}})$. Singularities develop when the area of $D$ shrinks to zero, which happens on the hyperplane $H$ in $B$.

  Calculations by the author, along the lines of §8 but more complicated, show that one can put together a fibration with the topological properties we want using only the local models of §5 and §7. There is no need to include any other kind of singular point.

The author is fairly confident about parts (a) and (c), but rather less happy about part (b). In fact, Ruan’s Lagrangian fibrations by gradient flow look quite like parts (a) and (c) in the relevant regions. Another option in part (b) is that there could be a new kind of codimension two singularity along the line segment $R_{1}\cap R_{2}\cap R_{3}$.

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.