Counting embedded curves in symplectic 6-manifolds

Abstract

Based on computations of Pandharipande, Zinger proved that the Gopakumar-Vafa BPS invariants $\mathrm{BPS}_{A,g}(X,ω)$ for primitive Calabi-Yau classes and arbitrary Fano classes $A$ on a symplectic $6$-manifold $(X,ω)$ agree with the signed count $n_{A,g}(X,ω)$ of embedded $J$-holomorphic curves representing $A$ and of genus $g$ for a generic almost complex structure $J$ compatible with $ω$. Zinger's proof of the invariance of $n_{A,g}(X,ω)$ is indirect, as it relies on Gromov-Witten theory. In this article we give a direct proof of the invariance of $n_{A,g}(X,ω)$. Furthermore, we prove that $n_{A,g}(X,ω) = 0$ for $g \gg 1$, thus proving the Gopakumar-Vafa finiteness conjecture for primitive Calabi-Yau classes and arbitrary Fano classes.