Counting embedded curves in symplectic 6-manifolds
Aleksander Doan, Thomas Walpuski
TL;DR
The authors develop a direct, self-contained construction of invariants counting embedded J-holomorphic curves in closed symplectic 6-manifolds and prove their invariance, including a genus bound that yields GV finiteness for primitive Calabi–Yau and Fano classes. The work hinges on a rigorous analysis of nodal J-holomorphic maps, their deformations, Gromov compactness, and a two-step Kuranishi model that treats ghost components and neck regions explicitly. By verifying a direct correspondence with Gopakumar–Vafa BPS invariants in the primitive CY case and extending to Fano classes under incidence constraints, the paper provides a comprehensive, geometry-driven proof of GV finiteness and integrality aspects. The methodology—versally deforming nodal curves, smoothing maps, and leveraging ghost-component obstruction analysis—offers a robust framework for future explorations of curve counts in higher-dimensional symplectic geometries and their GV-type formulas.
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.