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.