We shall proceed by giving a series of assumptions that define the properties of the fibrations we seek to construct. Here is the first, largely concerned with the symmetries of the fibration.

Really we would like the domain $U$ of $f$ to be all of $\mathbin{\mathbb{C}}^{3}$, and perhaps also to impose some asymptotic conditions on $f$ at infinity. But this would make our assumptions unnecessarily strong, and the author is not sure what asymptotic conditions would be appropriate. So instead we just suppose that $f$ is defined near $(0,0,0)$. We will not worry very much about the issues raised by $f$ not being defined on all of $\mathbin{\mathbb{C}}^{3}$, as they are primarily notational.

To understand where this list of properties has come from, note that the fibrations of Corollary 4.2 and Theorems 5.2 and 5.4 satisfy all of Assumption 7.1, except part (viii). We are aiming for a fibration that at a generic singular point is modelled one of the fibrations of Theorems 5.2 and 5.4, but will also have features in common with that of Corollary 4.2.

Therefore we simplify things by assuming that nearly all the symmetries these three fibrations have in common are also symmetries of the fibration we are aiming to construct. Here is our second assumption, drawing on the ideas of §6.

Here equation (54) essentially says that the fibres of $f$ may be written in the form (30). The explicit dependence on $c$ follows from part (v) of Assumption 7.1. Parts (i) and (ii) come from Proposition 6.1, and parts (iii)–(vi) from parts (i) and (vi)–(viii) of Assumption 7.1 respectively. Thus, the only thing Assumption 7.1 adds to Assumption 7.1 is that the fibres of $f$ may be written in the form (30).

Next we impose some conditions of a general topological nature which specify the ‘shape’ of the fibration we want, in particular the location and nature of its singularities.

We will see in §7.2 that $u_{a,b}$ and $v_{a,b}$ actually depend only on the values of $u_{a,b}$ on the $x$-axis. In Figure 1 we sketch the functions $u_{0,b}(x,0)$ for several values of $b$, on the same graph, to display the general features we expect of these functions. The curves are smooth except where they intersect the $x$-axis. The condition that $u_{0,b}(x,0)<u_{0,b^{\prime}}(x,0)$ when $b<b^{\prime}$ corresponds to the fact that the curves do not intersect, and move up the graph as $b$ increases.

For $\beta>0$ and $b=\beta^{2}$, the curve $u_{0,b}(x,0)$ intercepts the $x$-axis at $(\pm\beta,0)$. By part (d) of Propositions 6.7 and 6.8 and part (iii) of Assumption 7.1, near $(-\beta,0)$ we have $u_{0,b}(x,0)\approx(x+\beta)|x+\beta|$, and near $(\beta,0)$ we have $u_{0,b}(x,0)\approx-(x-\beta)|x-\beta|$. Thus $u_{0,b}(x,0)$ is differentiable at $(\pm\beta,0)$ with gradient zero.

For comparison, in Figure 2 we sketch the functions $u_{a,b}(x,0)$ for some small fixed $a\neq 0$, and the same values of $b$. The general shapes of the curves are the same, as are the intercepts with the $x$-axis. But the curves are all smooth, and using part (d) of Propositions 6.7 and 6.8 and part (iii) of Assumption 7.1, we see that

so that the gradient at $(-\beta,0)$ is approximately $|a|^{1/2}$, and at $(\beta,0)$ approximately $-|a|^{1/2}$. However, this approximation breaks down near $\beta=0$.

Recall that our goal is to model a fibration in which generic singular fibres in codimension one have two singularities, each locally a $T^{2}$-cone, but in codimension two these two cone points come together and fuse to form a new kind of singularity.

Assumption 7.1 implies that when $b=\beta^{2}$ for $\beta>0$, the fibre $N_{0,b,c}$ will have two singular points at $(0,0,\pm\beta+ic)$. Near $(0,0,-\beta+ic)$ it is locally modelled on the special Lagrangian $T^{2}$-cone $N_{0,-\beta+ic}$ of Definition 5, and near $(0,0,\beta+ic)$ it is locally modelled on the SL $T^{2}$-cone $N^{\prime}_{0,\beta+ic}$ of Definition 5.

When $b=0$, the fibre $N_{0,0,c}$ has one singular point at $(0,0,ic)$. It results from an isolated singular point of order 2 in $u_{0,0},v_{0,0}$, so as in §6.4 we expect the tangent cone at $(0,0,ic)$ to be the union of two special Lagrangian 3-planes $\Pi_{\pm}$ intersecting in a real line. We cannot describe the singularity of $N_{0,0,c}$ much more explicitly without proving Conjecture 6.14.

When $b<0$, the fibre $N_{0,b,c}$ is nonsingular. Thus the picture is that as $b$ decreases from positive to negative, two $T^{2}$-cone singular points in $N_{0,b,c}$ come together, fuse to form a new kind of singularity, and then vanish. We can think of the two singular points in $N_{0,b,c}$ for $b>0$ as having opposite sign, so that when they come together they cancel out.

We can now formulate our main conjecture.

The author is unable to prove Conjecture 7.4 at present, but here are some reasons for believing it, amounting to a sketch of a partial proof. In §7.1 we started with the fibration $f:U\rightarrow\mathbin{\mathbb{R}}^{3}$, and extracted the functions $u_{a,b},v_{a,b}$ from it. However, any proof of Conjecture 7.4 will probably go the other way, and first construct functions $u_{a,b},v_{a,b}$ satisfying (57), and then define the fibres $N_{a,b,c}$ by (54), and put them together to form $f$.

It is not obvious that if we did define families of functions $u_{a,b},v_{a,b}$ satisfying (57), then the corresponding SL 3-folds $N_{a,b,c}$ would actually be the fibres of a fibration. We will show that part (ii) of Assumption 7.1 comes close to ensuring that they do, because it implies that the $N_{a,b,c}$ are disjoint.

A 3-dimensional family of disjoint 3-folds in $\mathbin{\mathbb{C}}^{3}$ must locally define a fibration. So if we define $U$ to be the total space of all the $N_{a,b,c}$ then we do have a fibration $f:U\rightarrow\mathbin{\mathbb{R}}^{3}$ with fibres $N_{a,b,c}$. Therefore, we have more-or-less reduced the problem to finding families of functions $u_{a,b},v_{a,b}$, which need only be defined near $(0,0)$ in $\mathbin{\mathbb{R}}^{2}$ for small $a,b$, satisfying Assumption 7.1 and parts (i)–(vi) of Assumption 7.1.

Now by Proposition 6.4, given any real analytic values for $u_{a,b}$ and $v_{a,b}$ on the $x$-axis, there exist unique solutions of (57) near the $x$-axis with these values, except when $a=0$ near points $(x,0)$ with $u_{0,b}(x,0)=0$. But part (v) of Assumption 7.1 gives $v_{a,b}(x,0)\equiv 0$. Thus the function $u_{a,b}(x,0)$ captures all the essential information about the behaviour of $u_{a,b}$ and $v_{a,b}$ near the $x$-axis.

Note also that part (ii) of Assumption 7.1 can be restricted to the $x$-axis. For if $u_{a,b}(x,0)<u_{a,b^{\prime}}(x,0)$ for all $a,b,b^{\prime},x$ with $b<b^{\prime}$, then by continuity of the $u_{a,b}$ we have $u_{a,b}(x,y)<u_{a,b^{\prime}}(x,y)$ for sufficiently small $y$. Thus part (ii) holds near the $x$-axis, so by making $U$ smaller if necessary we ensure that part (ii) of Assumption 7.1 holds on all of $V_{a,b}$.

We may therefore try to proceed as follows. We choose real analytic functions $u_{a,b}(x,0)$ satisfying the applicable parts of Assumptions 7.1 and 7.1, in a fairly arbitrary way. We then extend these uniquely to satisfy (57) in a small open neighbourhood of the $x$-axis. Finally we construct the associated SL 3-folds, and argue that they locally form a fibration of $\mathbin{\mathbb{C}}^{3}$.

The main problem with this approach is when $a=0$ near the singular points $(x,0)$ with $u_{0,b}(x,0)=0$, as Proposition 6.4 does not apply. To overcome this problem we will need a good understanding of how solutions of (32) behave near isolated singular points of order 1 and 2. If we could prove Conjecture 6.14, then probably we could prove Conjecture 7.4 too.

We now discuss the holomorphic discs with boundary in the nonsingular fibres $N_{a,b,c}$, and their relation with the singularities of the singular fibres. For generic $a,b,c$ we expect all holomorphic discs in $\mathbin{\mathbb{C}}^{3}$ with boundary in $N_{a,b,c}$ to be $\mathbin{\rm U}(1)$-invariant, as the virtual dimension of the moduli space of such discs is zero. In the next proposition we identify all such holomorphic discs.

Now let $\hat{D}$ be a $\mathbin{\rm U}(1)$-invariant holomorphic disc in $\mathbin{\mathbb{C}}^{3}$ with boundary in $N_{a,b,c}$, and let $(z_{1},z_{2},z_{3})\in\hat{D}$. We claim that $z_{1}=0$ or $z_{2}=0$. Suppose $z_{1},z_{2}\neq 0$. As $D$ contains the $\mathbin{\rm U}(1)$-orbit of $(z_{1},z_{2},z_{3})$ and is holomorphic, it must locally contain the orbit of $(z_{1},z_{2},z_{3})$ under the complexification of the $\mathbin{\rm U}(1)$-action (52). Therefore, $D$ must locally be a subset of

But there are no $\mathbin{\rm U}(1)$-invariant holomorphic discs in this set; the best one can do is an annulus. Thus one of $z_{1},z_{2}$ must be zero.

Let $(z_{1},z_{2},z_{3})$ be a point in the boundary of $\hat{D}$. Then $(z_{1},z_{2},z_{3})\in N_{a,b,c}$, so $|z_{1}|^{2}-|z_{2}|^{2}=a>0$. This implies that $z_{1}\neq 0$, so $z_{2}=0$. It is then easy to show that $\hat{D}$ must be $\bigl\{(z,0,z_{3}):|z|^{2}\leqslant a\bigr\}$. This agrees with (58) with $x=\mathop{\rm Re}(z_{3})$ and $c=\mathop{\rm Im}(z_{3})$, and $(z_{1},0,z_{3})\in N_{a,b,c}$ implies that $u_{a,b}(x,0)=0$. So all $\mathbin{\rm U}(1)$-invariant holomorphic discs $\hat{D}$ with boundary in $N_{a,b,c}$ are as in the proposition. For $N_{-a,b,c}$ the argument works in the same way. $\square$

Now Assumption 7.1 determines all the zeros of the functions $u_{a,b}(x,0)$ exactly. When $b>0$, there are two zeros at $x=\pm\sqrt{b}$, when $b=0$ there is one zero at $x=0$, and when $b<0$ there are no zeros at all. Therefore the proposition shows that for $a\neq 0$, when $b>0$ there are two holomorphic discs with boundary in $N_{a,b,c}$, when $b=0$ there is one, and when $b<0$ there are none.

For generic $(a,b,c)$, these should be all the holomorphic discs with boundary in $N_{a,b,c}$. We think of the two holomorphic discs with boundary in $N_{a,b,c}$ for $b>0$ as having opposite sign. As $b$ decreases though zero, they come together and cancel out.

Such holomorphic discs are relevant to the singularities of special Lagrangian fibrations, for the following reason. Let $D$ be a holomorphic disc in a Calabi–Yau 3-fold $X$, with boundary in a special Lagrangian 3-fold $N$. Then the area of $D$ is $\int_{D}\omega=[\omega]\cdot[D]$, where $[\omega]$ is the relative de Rham cohomology class of $\omega$ in $H^{2}(X,N;\mathbin{\mathbb{R}})$, and $[D]$ the relative homology class of $D$ in $H_{2}(X,N;\mathbin{\mathbb{Z}})$.

Thus the area of $D$ depends only on its relative homology class, and homologous holomorphic discs have the same area. Clearly, the area of a holomorphic disc $D$ must be positive. So what happens when we deform $N$ so that the area $[\omega]\cdot[D]$ becomes zero? It turns out that usually $D$ shrinks to a point, and $N$ becomes singular. The singularity is the result of collapsing the boundary ${\mathcal{S}}^{1}$ of $D$ in $N$ to a point, and thus is a $T^{2}$-cone.

This has three important consequences:

• •

  The singularities induced by holomorphic discs shrinking to zero appear in codimension one in the base $B$ of an SL fibration $f:X\rightarrow B$, on the hyperplane where the area of the disc shrinks to zero.

• •

  There may be several homologous holomorphic discs $D_{1},\ldots,D_{k}$ with boundary in a generic fibre $N$. As the area of the discs shrinks to zero, $N$ will simultaneously develop $k$ singular points.

• •

  We can guess what the singularity looks like for generic singular fibres. The author is developing (and hopes one day to publish) a theory of singularities of SL 3-folds and their deformations. It predicts that when a holomorphic disc shrinks to zero, in the generic case the singularity that develops is locally modelled on the (isomorphic) $T^{2}$-cones $L_{0}^{\pm}$ of (16).

These three ideas were part of the author’s motivation in constructing the fibrations described above.