Counting embedded curves in symplectic $6$-manifolds

Abstract

Based on computations of Pandharipande (1999), Zinger (2011) proved that the Gopakumar–Vafa BPS invariants ${\mathrm{BPS}}_{A,g}(X,\omega )$ for primitive Calabi–Yau classes and arbitrary Fano classes $A$ on a symplectic $6$-manifold $(X,\omega )$ agree with the signed count $n_{A,g}(X,\omega )$ of embedded $J$-holomorphic curves representing $A$ and of genus $g$ for a generic almost complex structure $J$ compatible with $\omega$. Zinger's proof of the invariance of $n_{A,g}(X,\omega )$ is indirect, as it relies on Gromov–Witten theory. In this article we give a direct proof of the invariance of $n_{A,g}(X,\omega )$. Furthermore, we prove that $n_{A,g}(X,\omega )=0$ for $g\gg 1$, thus proving the Gopakumar–Vafa finiteness conjecture for primitive Calabi–Yau classes and arbitrary Fano classes.