Counting embedded curves in symplectic 6-manifolds

Abstract

Based on computations of Pandharipande, Zinger proved that the Gopakumar-Vafa BPS invariants ${\mathrm{BPS}}_{A,g}(X,\omega )$ 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.

References in corpus (16)

• Curve counting via stable pairs in the derived category
• Stable pairs and BPS invariants
• Hodge integrals and degenerate contributions
• BPS states of curves in Calabi--Yau 3--folds
• M-Theory and Topological Strings--I
• Relative Lefschetz Action and BPS State Counting
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• Castelnuovo's bound and rigidity in almost complex geometry
• Pseudohomology and homology
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• The Gopakumar-Vafa finiteness conjecture
• Skeins on Branes
• Complex surfaces and interpolation on pseudo-holomorphic cylinders
• Equivariant Brill-Noether theory for elliptic operators and super-rigidity of $J$-holomorphic maps

Cited by in corpus (4)

• Castelnuovo's bound and rigidity in almost complex geometry
• The Gopakumar-Vafa finiteness conjecture
• Open/Closed BPS Correspondence and Integrality
• Degree One Contributions and Open Gromov-Witten Invariants