Let $(M,J,g,\Omega)$ be a Calabi–Yau $m$-fold, either compact or suitably convex at infinity, and $D^{b}{\mathbin{\mathscr{F}}}(M)$ the derived Fukaya category of $M$ in the sense of [18, 20]. Then there exists a natural Bridgeland stability condition $(Z,{\mathbin{\cal P}})$ on $D^{b}{\mathbin{\mathscr{F}}}(M)$ such that:

• (a)

  The central charge $Z$ is the composition of the natural maps

  (3.1)

• (b)

  If $(L,E,b)\in D^{b}{\mathbin{\mathscr{F}}}(M)$ with $L$ special Lagrangian of phase $e^{i\pi\phi},$ so that $L$ has constant phase function $\theta_{L}=\pi\phi,$ then $(L,E,b)\in{\mathbin{\cal P}}(\phi)$.

• (c)

  (Dubious, probably false as stated.) Suppose we enlarge the definition of $D^{b}{\mathbin{\mathscr{F}}}(M)$ so that it contains ‘as many Lagrangians $L$ as possible for which $HF^{*}$ can be defined’, including immersed Lagrangians as in §2.6, and some classes of singular Lagrangians. Then every isomorphism class of objects in ${\mathbin{\cal P}}(\phi)$ for any $\phi\in{\mathbin{\mathbb{R}}}$ contains a unique representative $(L,E,b)$ with $L$ a (possibly immersed or singular) special Lagrangian of phase $e^{i\pi\phi}$.

Part (c) requires the inclusion of badly singular Lagrangians in $D^{b}{\mathbin{\mathscr{F}}}(M),$ which may not be feasible. Here is an alternative which may work with $D^{b}{\mathbin{\mathscr{F}}}(M)$ containing only more mildly singular Lagrangians:

• (c$\boldsymbol{)}{}^{\prime}$

  (Still dubious.) Suppose we enlarge $D^{b}{\mathbin{\mathscr{F}}}(M)$ so that it contains ‘sufficiently many Lagrangians $L$ for which $HF^{*}$ can be defined’, including immersed and some singular Lagrangians. Then for any $\epsilon>0$ and $\phi\in{\mathbin{\mathbb{R}}},$ every isomorphism class of objects in ${\mathbin{\cal P}}(\phi)$ contains a representative $(L,E,b)$ whose phase function $\theta_{L}$ maps $\theta_{L}:L\rightarrow(\pi\phi-\epsilon,\pi\phi+\epsilon)$.