Any holomorphic sheaf can be twisted by a sufficiently positive line bundle $\mathscr{O}(N)$ so that it has sections; equivalently there are homomorphisms to the bundle from any sufficiently negative line bundle. If the sheaf has global support, this homomorphism is injective, exhibiting $E$ as an extension

One test of our notion of subobject of Lagrangians (in terms of connect sums), then, is that there should be appropriate connect sums mirroring this extension.

A line bundle $\mathcal{L}$ defines a spherical object [ST] of the derived category of sheaves on a Calabi-Yau manifold $X$; that is Ext${}^{i}(\mathcal{L},\mathcal{L})=H^{0,i}(X)\cong H^{*}(S^{n};\mathbb{C}\,)$ is $\mathbb{C}\,$ in dimensions $0$ and $n$, and zero otherwise. These should be mirror to Lagrangian homology spheres; we will consider only spheres here so that we can use the graded Dehn twists [S2] around them. Negativity compared to some other Lagrangian may not make sense in general (intuitively, the Lagrangian might be mirror not to a sheaf but to an object of the derived category with Homs in negative degrees, etc.) but instead we can consider only those spheres $L$ with only degree zero intersection points (3.4) with a fixed Lagrangian $L^{\prime}$.

Then it is indeed true that we can exhibit $L$ as a subobject of $L^{\prime}$: denoting by $T_{L}$ the (graded) symplectic Dehn twist about $L$, simply note that

expresses $L^{\prime}$ as a connect sum of $L$ and something else. These relations can be shown by grading similar results in [S1]. In general this will not destabilise $L^{\prime}$ due to the phase of $L$ being so negative.

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

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

(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

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

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:

In order to understand limits of mean curvature flow it will be useful to have the following concept; an analogue for Lagrangians of the Jordan-Hölder filtration of sheaves (see [HuL] 1.5, for instance).

In sheaf theory the Jordan-Hölder filtration need not be unique, but the decomposition is. For smooth connected Lagrangians, with connected $L_{i}$ for all $i$, however, we expect the filtration to be unique too; the difference is essentially that while direct sum is an operation on bundles, we are proposing that its mirror is the (singular) union of Lagrangians, and this cannot give a smooth Lagrangian if there is non-zero Floer cohomology between the two Lagrangians.

If we assume the conjecture of [Th] and Section 7, and the properties of Floer cohomology [FO3] for all of the above Lagrangians (e.g. if they are homology spheres), we can demonstrate the existence and uniqueness of the Jordan-Hölder filtration for a Lagrangian $L$ whose phase function of $L$ satisfies $\sup\theta_{L}-\inf\theta_{L}<\pi$.

Without loss of generality we may assume (by rotating $\Omega$) that $\theta$ lies between $\pi/2-\epsilon$ and $-\pi/2+\epsilon$, for some $\epsilon>0$. By the inequality (5.2) above, then, any $L_{1}\#L^{\prime}$ destabilising it will satisfy $\phi(L_{1}),\phi(L^{\prime})\in(-\pi/2+\epsilon,\pi/2-\epsilon)$.

We choose such an $L_{1}$ of maximal phase, and, amongst other such $L_{1}$s of the same phase, minimal $\int_{L_{1}}\mathrm{Re}\,\Omega$ (for the purposes of this proof we will call this quantity cohomological volume). This still need not specify $L_{1}$ uniquely though.

We claim that such an $L_{1}$ must be stable by construction. Any subobject of $l\leq L_{1}$ would also be a subobject of $L$ and so by the construction of $L_{1}$ must either have smaller phase, which is not possible since it destabilises $L_{1}$, or equal phase and greater or equal cohomological volume. But $L_{1}=l\#l^{\prime}$, where $l^{\prime}=L_{1}/l$ has phase $\phi(l^{\prime})=\phi(L_{1})\in(-\pi/2,\pi/2)$ and so positive cohomological volume $\int_{l^{\prime}}\mathrm{Re}\,\Omega$. So the complex numbers

all have positive real part, so that the cohomological volume of $l$ is strictly less that that of $L_{1}$, a contradiction.

We then apply the same procedure to $L^{\prime}$, producing an $L_{2}\hookrightarrow L^{\prime}$, and so on. By construction $\phi(L^{\prime}/L_{2})\leq\phi(L^{\prime})\leq\phi(L)<\pi/2-\epsilon$, and there is a canonical morphism (5.1) in $HF^{0}(L,L^{\prime}/L_{2})\neq 0$, making $\phi(L^{\prime}/L_{2})\geq\inf_{L}\theta_{L}>-\pi/2+\epsilon$ by (5.2).

Thus, inductively, we get the same inequalities at each stage, and the cohomological volume of $L^{\prime}$ decreases strictly with each decomposition $L\approx L_{1}\#\ldots\#L_{n}\#L^{\prime}$. The cohomological volume of any $l$ with phase $\phi(l)\in(-\pi/2+\epsilon,\pi/2-\epsilon)$ is greater than (or equal to in the SLag case) $\cos(\pi/2-\epsilon)\int_{l}\vol$, by (1.1), where $\vol$ is its Riemannian volume form. This is bounded below above zero, so the process can have at most a finite number of steps.

This gives us the Jordan-Hölder filtration; next we consider uniqueness when the $L_{i}$s are connected (assuming the conjectures of [Th] and Section 7 and some Floer cohomology). Suppose that $L_{1}^{\prime}\leq L_{2}^{\prime}\leq\ldots\leq L$ is another such connected decomposition. If $HF^{0}(L_{1}^{\prime},L_{1})\neq 0$ then by the proof of Theorem 4.3 (which applies as $L_{1}^{\prime}$ and $L_{1}$ have the same phase), $L_{1}$ and $L_{1}^{\prime}$ are equal, and we pass to $L_{2}$.

If, however, $HF^{0}(L_{1}^{\prime},L_{1})=0$, then we claim that $HF^{0}(L_{1}^{\prime},L/L_{1})\neq 0$. Again this should follow from standard facts about Floer cohomology, in particular a long exact sequence $HF^{*}(L_{1}^{\prime},L_{1})\to HF^{*}(L_{1}^{\prime},L)\to HF^{*}(L_{1}^{\prime},L/L_{1})\to HF^{*+1}(L_{1}^{\prime},L_{1})$. For $L/L_{1}$ a sphere this is Seidel’s exact sequence ([S3] Theorem 3.3), and in general one can establish it at the level of chains by good choices of hamiltonian perturbations as in Section 5.2; as usual the problem is in controlling the differential, i.e. holomorphic discs.

Assuming this we may pass to $L_{2}$; inductively we eventually obtain that $L_{1}^{\prime}$ is isomorphic to one of the graded pieces $L_{i+1}/L_{i}$ of the original filtration, and is a subobject of $L_{i+1}$ but not of $L_{i}$. But this gives us a contradiction (in contrast to the sheaf analogue), since we have that both $L_{i+1}\approx L_{i}\#L_{1}^{\prime}$ and $L_{1}^{\prime}$ is a subobject of $L_{i+1}$. The first condition ensures that there are representatives of the hamiltonian deformation classes such that $L_{i+1}$ and $L_{1}^{\prime}$ have no index $n$ intersection points by the construction of (5.1), so that $HF^{n}(L_{i+1},L_{1}^{\prime})=0$. But this is $HF^{0}(L_{1}^{\prime},L_{i+1})^{*}$, which cannot vanish by the second condition. (It is here we use the connectivity condition, i.e. that the connect sum $L_{i+1}=L_{1}^{\prime}\#(L_{i+1}/L_{1}^{\prime})$ is not a trivial disjoint union. Without the connectivity condition the usual proof (e.g. [HuL] 1.5) the the Jordan-Hölder decomposition (rather than filtration) of sheaves is unique applies to Lagrangians, now that we have proved or assumed all (the mirror analogues of) the algebraic facts used for sheaves in terms of Floer cohomology instead of Exts.)

So in the simplest case of instability, such as the example of Joyce considered in [Th], where $L=L_{1}\#L_{2}$ is the only relevant decomposition of $L$ with $\phi(L_{1})\geq\phi(L_{2})$, the Jordan-Hölder decomposition (5.4) would be simply $L_{1}\cup L_{2}$ (where the $L_{i}$ are SLag representatives of their classes). This, like all such decompositions, is in the closure of the hamiltonian deformation orbit of $L$ while not being in the orbit itself.

This should have relevance to the Schoen-Wolfson programme [SW1], [SW2] to find canonical representatives (in a fixed hamiltonian deformation class) of Lagrangian homology classes using volume minimisers and so SLags (they do not use a flow, but regularity results to study minimising currents). Our conjecture (as in [Th] and later in Section 7) should either provide a unique SLag in a hamiltonian deformation class, or a number of SLags in a Jordan-Hölder decomposition.

For instance in the example above of $L_{1}\#L_{2}$ in 2 dimensions (where their programme has been worked out [SW2]) we would produce SLags in the classes of $L_{1}$ and $L_{2}$, but we could also form $L_{2}\#L_{1}$; this could then be stable (it is no longer destabilised by either of the $L_{i}$; if the phases of the $L_{i}$ are sufficiently close one can show that in fact nothing else destabilises it either) and we should recover a SLag in this class (and so in the same homology class in two dimensions).

Since in two dimensions SLags are just holomorphic curves with respect to a different complex structure, this places heavy restrictions on stability. Take the $L_{i}$ above to be spheres in $K3$ surfaces. Then any holomorphic sphere is unique in its homology class (it has negative self intersection $-2$, so does not lie in a pencil). Any other hamiltonian deformation class must therefore be unstable. Good examples are provided by taking a stable (SLag/holomorphic) sphere, and applying the square of a Dehn twist $T^{2}_{L_{1}}$ to it; this preserves homology classes but can change hamiltonian deformation classes. If it does it should produce an unstable Lagrangian with copies of $L_{1}$ in its Jordan-Hölder decomposition; this happens in all simple cases. $L_{1}\#L_{2}$ is taken to $L_{2}\#L_{1}$, for instance; only one of these can be stable, the other having a Jordan-Hölder decomposition $L_{1}\cup L_{2}$ in the simplest case.

More generally, instead of studying the action on individual (S)Lags of symplectomorphisms like $T^{2}_{L_{1}}$ above, we could try to study them all at once by studying the Lagrangian graph of the symplectomorphism in $X\times X$, and its mean curvature flow. This looks for minimal energy representatives of the hamiltonian isotopy class of a symplectomorphism, and breaks graphs up into correspondences representing singular maps (birational maps in the hyperkähler case) with singularities concentrated in loci whose stability is affected by the symplectomorphism. For a Dehn twist $T_{L}$, for instance, we would expect to get the graph $\Delta$ of the identity union $L\times L$. This also shows what the analogue of a Dehn twist $T_{L}$ should be when $L$ is not a sphere but a rational homology sphere (so that it is still spherical to complex coefficients, and so mirror to a twist on the derived category of sheaves on the mirror Calabi-Yau [ST]). Namely $\Delta\cup L\times L$ is a Lagrangian correspondence in $X\times X$ which should give an automorphism of the derived Fukaya category of $X$ (by the usual Fourier-Mukai-type construction) not induced by a symplectomorphism of $X$.