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.

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  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.

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  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.