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

$$0\to\mathscr{O}(-N)\to E\to Q\to 0.$$

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

$$L^{\prime}\approx T_{L}^{-1}T_{L}L^{\prime}\approx L\#[L^{\prime}\#(L[\,1\,])]$$

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.