We temporarily write $D^{b}{\mathbin{\mathscr{F}}}(M)_{\rm nonsing}$ for the derived Fukaya category of nonsingular immersed Lagrangians, and $D^{b}{\mathbin{\mathscr{F}}}(M)_{\rm sing}$ for the category including Lagrangians with ‘stable special Lagrangian singularities’. It seems likely that $D^{b}{\mathbin{\mathscr{F}}}(M)_{\rm sing}$ and $D^{b}{\mathbin{\mathscr{F}}}(M)_{\rm nonsing}$ need not be equivalent categories. If so, $D^{b}{\mathbin{\mathscr{F}}}(M)_{\rm sing}$ may be preferable to $D^{b}{\mathbin{\mathscr{F}}}(M)_{\rm nonsing}$, in the sense of being better behaved, more natural, or the right category to use in Mirror Symmetry. To test this, we should start in dimension $m=3$ by including Lagrangians with isolated singularities modelled on the $T^{2}$-cone $C$ in (2.4).

The following example was suggested to me by Ivan Smith. Harris [25] constructs a smooth family $(M^{t},\omega^{t}):t\in[0,\epsilon)$ of symplectic Calabi–Yau 6-manifolds for small $\epsilon>0$, with the following properties:

• (i)

  $M^{t}$ is independent of $t$, and is the result of adding a 2-handle to $T^{*}{\mathbin{\cal S}}^{3}$. There is an isomorphism $H^{2}(M^{t},{\mathbin{\mathbb{R}}})\cong{\mathbin{\mathbb{R}}}$ identifying $[\omega^{t}]$ with $t$. Thus $(M^{t},\omega^{t})$ is an exact symplectic manifold if and only if $t=0$.

• (ii)

  For $t>0$ there is a compact, embedded Lagrangian $L^{t}$ in $(M^{t},\omega^{t})$ diffeomorphic to ${\mathbin{\cal S}}^{3}$, depending smoothly on $t$, with $0\neq[L^{t}]\in H_{3}(M^{t};{\mathbin{\mathbb{Z}}})\cong{\mathbin{\mathbb{Z}}}$.

• (iii)

  There are no Lagrangian ${\mathbin{\cal S}}^{3}$’s in $(M^{0},\omega^{0})$, and in fact, no compact, exact, embedded Lagrangians in $(M^{0},\omega^{0})$ at all.

• (iv)

  As in [25, Rem. 3.7], $L^{0}=\lim_{t\rightarrow 0}L^{t}$ is a singular Lagrangian in $M^{0}$, which topologically looks like an ${\mathbin{\cal S}}^{3}$ with an ${\mathbin{\cal S}}^{1}$ collapsed to a point $p$, so that topologically $L^{0}$ is modelled on a $T^{2}$-cone near $p$.

All this suggests that $D^{b}{\mathbin{\mathscr{F}}}(M^{t})_{\rm nonsing}$ is empty for $t=0$, and nonempty for $t>0$. This counts as pathological behaviour, discontinuous in $t$, since the $D^{b}{\mathbin{\mathscr{F}}}(M^{t})_{\rm nonsing}$ for small $t>0$ are not deformations of $D^{b}{\mathbin{\mathscr{F}}}(M^{0})_{\rm nonsing}$ in a meaningful sense. Intuitively, one would expect objects to disappear under small deformations owing to obstructions, so that $D^{b}{\mathbin{\mathscr{F}}}(M^{t})_{\rm nonsing}$ for $t>0$ should be smaller than $D^{b}{\mathbin{\mathscr{F}}}(M^{0})_{\rm nonsing}$.

It seems plausible that we can choose the $L^{t}$ up to Hamiltonian isotopy so that $L^{0}$ has one singular point $p$ locally modelled on $C$ in (2.4), and $L^{t}$ for $t>0$ is locally modelled near $p$ on $L_{1}^{A(t)}$ in (2.5), where $A(t)\rightarrow 0$ as $t\rightarrow 0$. If so, $L^{0}$ may give an object in $D^{b}{\mathbin{\mathscr{F}}}(M^{0})_{\rm sing}$, and the derived categories $D^{b}{\mathbin{\mathscr{F}}}(M^{t})_{\rm sing}$ may depend continuously on $t\in[0,\epsilon)$. So in this example, $D^{b}{\mathbin{\mathscr{F}}}(M)_{\rm sing}$ may be better behaved than $D^{b}{\mathbin{\mathscr{F}}}(M)_{\rm nonsing}$ under deformations of $(M,\omega)$.