The following behaviour, called ‘collapsing a zero object’, is a possible model for finite time singularities in the programme of §3.2.

Let $(M,J,g,\Omega)$ be a Calabi–Yau $m$-fold, and extend $D^{b}{\mathbin{\mathscr{F}}}(M)$ to include immersed Lagrangians, as in [2]. Suppose $\{(L^{t},E^{t}):t\in(T-\epsilon,T+\epsilon)\}$ for $\epsilon>0$ small is a family of Lagrangian branes in $M$ with $HF^{*}$ unobstructed, and $\{b^{t}:t\in(T-\epsilon,T+\epsilon)\}$ a corresponding family of bounding cochains, satisfying the following conditions:

• (i)

  The $(L^{t},E^{t},b^{t})$ for $t\in(T-\epsilon,T+\epsilon)$ are all isomorphic in $D^{b}{\mathbin{\mathscr{F}}}(M)$.

• (ii)

  When $t<T,$ $L^{t},E^{t}$ depend smoothly on $t\in(T-\epsilon,T),$ and $\{(L^{t},E^{t}):t\in(T-\epsilon,T)\}$ satisfies Lagrangian MCF, with a finite time singularity at $t=T,$ with one singular point $p\in M$.

  Similarly, when $t\geqslant T,$ $L^{t},E^{t}$ depend smoothly on $t\in[T,T+\epsilon),$ and $\{(L^{t},E^{t}):t\in[T,T+\epsilon)\}$ satisfies Lagrangian MCF.

• (iii)

  For $t\in(T-\epsilon,T)$ there is a decomposition $(L^{t},E^{t},b^{t})=(L^{t}_{1},E^{t}_{1},b^{t}_{1})\amalg(L^{t}_{2},E^{t}_{2},b^{t}_{2}),$ with $L^{t}_{1},L^{t}_{2}$ open and closed in $L^{t}$. There exists a continuous $\delta:(T-\epsilon,T)\rightarrow(0,\infty)$ with $\delta(t)\rightarrow 0$ as $t\rightarrow T$ such that $L^{t}_{1}\subseteq B_{\delta(t)}(p)$ for all $t\in(T-\epsilon,T),$ where $B_{\delta(t)}(p)$ is the open ball of radius $\delta(t)$ about $p$ in $M$. That is, the whole of $L^{t}_{1}$ converges uniformly to $p\in M$ as $t\rightarrow T$.

• (iv)

  $(L^{t}_{1},E^{t}_{1},b^{t}_{1})\cong 0$ in $D^{b}{\mathbin{\mathscr{F}}}(M)$ for $t\in(T-\epsilon,T),$ so that $(L^{t},E^{t},b^{t})\cong(L^{t}_{2},E^{t}_{2},b^{t}_{2})$ in $D^{b}{\mathbin{\mathscr{F}}}(M)$.

• (v)

  The family $\{(L^{t}_{2},E^{t}_{2},b^{t}_{2}):t\in(T-\epsilon,T)\}\amalg\{(L^{t},E^{t},b^{t}):t\in[T,T+\epsilon)\}$ is smooth in $t\in(T-\epsilon,T+\epsilon)$.

Rather than taking $L^{T}=L^{T}_{2}$ to be a nonsingular immersed Lagrangian at $t=T,$ we could instead write $L^{T}=\{p\}\amalg L^{T}_{2},$ where $\{p\}=\lim_{t\rightarrow T}L^{t}_{1}$ is regarded as an extreme example of a singular Lagrangian in $M$.