We now describe two very elementary examples of (smooth) special Lagrangian fibrations of $\mathbin{\mathbb{C}}^{3}$, which we will build on later. The results of this section are not new, and can mostly be found in Harvey and Lawson [9, §III.3] and the author [10, §3]. The proofs are easy and will generally be omitted. Here is our first family of SL 3-folds in $\mathbin{\mathbb{C}}^{3}$.

Clearly $K_{a,b,c}$ is invariant under the group $\mathbin{\rm U}(1)\times\mathbin{\mathbb{R}}$ acting on $\mathbin{\mathbb{C}}^{3}$ by

and using the methods of [11] one can show that any connected SL 3-fold in $\mathbin{\mathbb{C}}^{3}$ invariant under this group is a subset of some $K_{a,b,c}$. From the theorem we immediately deduce:

This fibration is the local model for the most generic kind of singularity in smooth SL fibrations of Calabi–Yau 3-folds, as studied by Gross [5, 6], for instance. Note that the set of singular fibres in $\mathbin{\mathbb{R}}^{3}$ is $\bigl\{(0,0,c):c\in\mathbin{\mathbb{R}}\bigr\}$, of codimension two, and each singular fibre has a one-dimensional singular set $\bigl\{(0,0,t+ic):t\in\mathbin{\mathbb{R}}\bigr\}$. Also, the set of all singular points of singular fibres is $\bigl\{(0,0,z_{3}):z_{3}\in\mathbin{\mathbb{C}}\bigr\}$, a complex line in $\mathbin{\mathbb{C}}^{3}$.

Here is our second family of SL 3-folds in $\mathbin{\mathbb{C}}^{3}$, due originally to Harvey and Lawson [9, §III.3.A].

Again, these 3-folds $L_{a,b,c}$ have a two-dimensional symmetry group, this time

which acts on $\mathbin{\mathbb{C}}^{3}$ as a subgroup of $\mathop{\rm SU}(3)$ by

Any connected SL 3-fold in $\mathbin{\mathbb{C}}^{3}$ invariant under this group is a subset of some $L_{a,b,c}$. The theorem immediately yields

This is the local model for another, nongeneric kind of singularity in smooth SL fibrations of Calabi–Yau 3-folds. The set of singular fibres in $\mathbin{\mathbb{R}}^{3}$ is

which is three half-lines meeting at a point, and is again of codimension two in $\mathbin{\mathbb{R}}^{3}$. Generic singular fibres have singular fibre a circle, which is one-dimensional. Note that in a small neighbourhood of a singular point of a generic singular fibre, the fibration is a smooth deformation of the fibration of Corollary 4.2. The set of all singular points of singular fibres is

a singular complex curve in $\mathbin{\mathbb{C}}^{3}$.

In the rest of the section we explore the structure of the singular fibres in cases (i)–(iv) of Theorem 4.3, following [10, §3].

So define $L_{1,\alpha}^{\pm}=\phi_{1,\alpha}^{\pm}({\mathcal{S}}^{1}\times\mathbin{\mathbb{C}})$. An equivalent definition is

Then $L_{1,\alpha}^{+}$ and $L_{1,\alpha}^{-}$ are both nonsingular, embedded 3-submanifolds of $\mathbin{\mathbb{C}}^{3}$ diffeomorphic to ${\mathcal{S}}^{1}\times\mathbin{\mathbb{C}}$. Comparing (12) and (17) we see that $L_{\alpha,\alpha,0}=L_{1,\alpha}^{+}\cup L_{1,\alpha}^{-}$. Since $L_{\alpha,\alpha,0}$ is an SL 3-fold we deduce that $L_{1,\alpha}^{\pm}$ are also SL 3-folds in $\mathbin{\mathbb{C}}^{3}$, which is easy to verify directly.

Observe that $L_{1,\alpha}^{+}\cap L_{1,\alpha}^{-}=\bigl\{(\alpha^{1/2}{\rm e}^{i\theta},0,0):\theta\in[0,2\pi)\bigr\}$, which is the singular set of $L_{\alpha,\alpha,0}$ given in Theorem 4.3. Thus $L_{\alpha,\alpha,0}$ is the union of two nonsingular special Lagrangian 3-folds $L_{1,\alpha}^{+}$ and $L_{1,\alpha}^{-}$, and the singularities of $L_{\alpha,\alpha,0}$ occur at their intersection. Note that we could consider $L_{\alpha,\alpha,0}$ to be a nonsingular, immersed submanifold.

It is not difficult to show that $L_{j,\alpha}^{+}$ is asymptotic to the $T^{2}$-cone $L_{0}^{+}$ at infinity for $j=1,2,3$. Thus the $L_{j,\alpha}^{+}$ for $j=1,2,3$ are three different families of asymptotically conical SL 3-folds asymptotic to the same singular cone $L_{0}^{+}$. We may interpret them as three different ways to ‘resolve’ the same SL 3-fold singularity $L_{0}^{+}$. This point of view was taken in [10, §3–§5]. Similarly, $L_{j,\alpha}^{-}$ is asymptotic to $L_{0}^{-}$ at infinity for $j=1,2,3$.

For $\alpha>0$, define a holomorphic disc $D_{1,\alpha}$ in $\mathbin{\mathbb{C}}^{3}$ by

Then $\partial D_{1,\alpha}$ is the intersection of $L_{1,\alpha}^{+}$ and $L_{1,\alpha}^{-}$. Therefore $D_{1,\alpha}$ is a holomorphic disc with boundary in both of the nonsingular SL 3-folds $L_{1,\alpha}^{\pm}$. In the same way, there are holomorphic discs $D_{2,\alpha}$ and $D_{3,\alpha}$ with boundaries in $L_{2,\alpha}^{\pm}$ and $L_{3,\alpha}^{\pm}$. This will be significant later, when we discuss holomorphic discs in Calabi–Yau 3-folds with boundary in special Lagrangian 3-folds.