Localizing a Dirac operator via $J$-holomorphic curves

Abstract

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 $$ \operatorname{ind}{\mathcal D}=3c_2(M)-c_1^2(M). $$ We then use the curve $C$ to construct a complex-linear perturbation ${\mathcal A}$ of ${\mathcal D}$ whose singular set is $Z_{\alpha}\sqcup C$, where $Z_{\alpha}=\alpha^{-1}(0)$ for a transverse section $\alpha$ of $\Lambda^{1,0}M$ nonvanishing on $C$. Applying Maridakis' index localization theorem, we express $\operatorname{ind}{\mathcal D}$ as the sum of localized contributions: $3c_2(M)$ from $Z_{\alpha}$ and $-c_1^2(M)$ from $C$.