It is usual in correspondences between stable objects in algebraic geometry and solutions of the corresponding moment map PDE for one direction of the correspondence to be reasonably straightforward to prove, namely that objects which satisfy the PDE are stable.

While we cannot prove this for SLags, we can show, for SLags satisfying Floer cohomology restrictions as in Theorem 4.3 (in particular for spheres), that they cannot be destabilised by other SLags. (To test for stability of sheaves it is sufficient to test only with stable subsheaves; if the conjecture of [Th] is true then similarly we could test for stability of Lagrangians by connect summing only SLags; this would then be enough to prove the general stability of SLags.)

The idea is that if $\phi(L_{1})>\phi(L)$, with both $L_{1}$ and $L$ SLags, then the Floer index of any intersection point of $L_{1}$ and $L$ is strictly positive (3.4), almost by definition. But if $L_{1}$ were to destabilise $L$, i.e. $L=L_{1}\#L_{2}$ for some $L_{2}$, then there should be canonical morphisms $HF^{0}(L_{1},L)\neq 0$ and $HF^{0}(L,L_{2})\neq 0$, a contradiction.

The morphism from $L_{1}\#L_{2}$ to $L_{2}$, by which we mean an element of

$$HF^{0}(L_{1}\#L_{2},L_{2}),$$

(5.1)

can be described as follows (the element of $HF^{0}(L_{1},L_{1}\#L_{2})$ is similar). We use the description of the connect sum in Section 3. Choose a Morse function $f$ on $L_{2}$ which has local maxima at intersection points $p$ with $L_{1}$, and in local Darboux charts as in Section 3, is pulled up from a function on $\gamma_{2}$. Let the function have a unique local minimum elsewhere on $L_{2}$, and now use this to hamiltonian deform $L_{2}$ off $L_{1}\#L_{2}$. By construction $L_{1}$ and $L_{1}\#L_{2}$ now intersect at the critical points of $f$ only, with Floer index the Morse index of $f$. In terms of Figure 1, as $f$ has a maximum on $\gamma_{2}$ at the vertex of $\gamma_{2}$, it defines a hamiltonian deformation of $\gamma_{2}$ downwards, away from the connect-sum neck. As $L_{2}$ only intersects $L_{1}$ near these connect-sum necks, we can make our charts small enough that $L_{2}$ now only intersects $L_{1}\#L_{2}$ where its hamiltonian deformation intersects the old $L_{2}$, i.e. at the critical points of $f$.

We now have a unique index zero point of $(L_{1}\#L_{2})\cap L_{2}$ at the unique local minimum of $f$. What we require is that this survives in the passage to cohomology of the cochain complex to give $HF^{0}(L_{2},L_{1}\#L_{2})\cong\mathbb{C}\,$. For instance, if there are no index one points (i.e. $f$ is a Morse function with only minima and index $\geq 2$ critical points) then this will clearly be the case. More generally there is a spectral sequence analogous to Poźniak’s [P] with

$$\mathrm{coker}\,\{\bigoplus_{i}\mathbb{C}\,_{p_{i}}[-n]\to H^{*}(L_{2})\}\ \Longrightarrow\ HF^{*}(L_{1}\#L_{2},L_{2})$$

(with a certain bigrading) converging to $HF^{*}(L_{1}\#L_{2},L_{2})$. Here the notation means that a copy of $\mathbb{C}\,$ is mapped to $H^{n}(L_{2})$ (i.e. it is in degree $n$) for every intersection point $p_{i}$ via the Morse theory for $f$ (whose maxima are at the $p_{i}$). Therefore the degree zero part also survives if, for instance, $H^{1}(L_{2})=0$. Another case we can deal with to get the same result is if $L_{2}$ is a sphere so that we can apply Seidel’s exact sequence [S3].

Using similar methods on Lagrangians rather than SLags, we can cut down on the number of possible destabilising Lagrangians $L_{1}$ we must check to conclude that a given $L$ is stable, rather analogously to only checking for subsheaves of vector bundles amongst those of lower rank. There are no morphisms (non-zero elements of $HF^{0}(L_{1},L)$) if the phase of $L_{1}$, at an intersection point $p$, is greater than that of $L$; the Floer index at $p$ is strictly positive. So for $L_{1}$ to destabilise $L$ (and so $HF^{0}(L_{1},L)\neq 0$ for Lagrangians satisfying the same conditions as above and in (4.3), e.g. homology spheres) we must have

$$\inf_{L_{1}}\theta_{L_{1}}<\sup_{L}\theta_{L},$$

and in fact the corresponding phase inequality at each point of intersection.

Thus we do not have to check all Lagrangians $L_{1},\,L_{2}$ for the stability of $L^{\prime}$ in [Th], just those whose phase function satisfies

$$\inf_{L_{1}}\theta_{L_{1}}\leq\sup_{L}\theta_{L}\hskip 10.00002pt\text{and}\hskip 10.00002pt\sup_{L_{2}}\theta_{L_{2}}\geq\inf_{L}\theta_{L},$$

where we can in fact replace the left hand sides of these inequalities by the sup (inf) over all Lagrangians in the same hamiltonian deformation class respectively.

Assuming the conjecture in [Th], so that we need only check SLag destabilisers, this reduces checking for destabilising subobjects amongst those Lagrangians satisfying the following cohomological conditions:

$$\inf_{L}\theta_{L}\leq\phi(L_{1})\leq\phi(L_{2})\leq\sup_{L}\theta_{L}.$$

(5.2)