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 $R_{1}\cap R_{2}\cap R_{3}$.