We now comment on the gap between what we proved and the (weak version of) automatic transversality that we will later assume.

1. 1.

   Prop. 3.8, Lemma 3.5 and Cor. 3.6 establish the dichotomy for holomorphic discs $u:\Sigma\to X$ arising in virtual dimension $n-1$ moduli spaces, that either $u$ is automatically transverse, or $\Omega(\cdot,v_{1},\ldots v_{n-1})$ vanishes for any $(n-1)$ first order deformation vectors. This argument does not establish unperturbed regularity for the lower dimensional moduli spaces, so it is not completely clear if complex structure perturbations can be removed in the arguments for $\partial\mathcal{C}=L-L^{\prime}$ in section 2.9.

2. 2.

   For the bad curves, $\Omega(\cdot,v_{1},\ldots v_{n-1})$ vanishes identically as a 1-form on $\Sigma$, so at any point on the boundary, $v_{1},\ldots v_{n-1}$ and the tangent vector to $\partial\Sigma$ are $\mathbb{R}$-linearly dependent. Suppose for the moment that the moduli spaces are regular, then the boundary evaluation to $L\cup L^{\prime}$ for the bad curves arise in Hausdorff dimension at most $n-1$. Morever, since the Solomon functional is defined through $\int_{\mathcal{C}}\lambda\wedge\Omega$, and $\Omega$ vanishes around the bad curves, smoothness assumptions imply that the bad curves cannot contribute.

   When regularity assumptions are dropped, one needs to appeal to virtual techniques, so these conclusions require further justification. One problem is that the standard virtual perturbation techniques based on Kuranishi structures do not necessarily produce virtual cycles inside the original moduli spaces, but only inside their small neighbourhoods. This perturbation step destroys the identical vanishing of $\Omega$, by a small amount corresponding to the size of the perturbation. As one shrinks the size of the perturbations, one needs uniform mass bound on the virtual chains to justify that the integral contribution to $\int_{\mathcal{C}}\lambda\wedge\Omega$ from the bad curves actually converges to zero.

3. 3.

   Alternatively, one can hope to replace Lagrangians by arbitrarily small Hamiltonian perturbations to achieve transversality. This is mostly adequate for our purpose, except that one needs to justify the ‘somewhere boundary injectivity’ property (cf. Remark 3.8).