Localizing a Dirac operator via $J$-holomorphic curves

Let $(M,J)$ be a compact almost hermitian $4$-manifold with a smooth embedded $J$-holomorphic curve $C$ representing the canonical class. Motivated by the symplectic Bogomolov–Miyaoka–Yau conjecture, we choose a twisted spin${}^{\text{c}}$ Dirac operator $\mathcal{D}$ on $M$ satisfying $$\mathrm{ind}\mathcal{D}=3c_2(M)-c_1^2(M)\mathrm{.}$$ We then use the curve $C$ to construct a complex-linear perturbation $\mathcal{A}$ of $\mathcal{D}$ whose singular set is $Z_{\alpha }\bigsqcup C$, where $Z_{\alpha }={\alpha }^{-1}(0)$ for a transverse section $\alpha$ of ${\mathrm{\Lambda }}^{1,0}M$ nonvanishing on $C$. Applying Maridakis’ index localization theorem, we express $\mathrm{ind}\mathcal{D}$ as the sum of localized contributions: $3c_2(M)$ from $Z_{\alpha }$ and $-c_1^2(M)$ from $C$.