Counting embedded curves in symplectic 6-manifolds

Abstract

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

References in corpus (1)

• Skeins on Branes