Counting embedded curves in symplectic 6-manifolds
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arXiv
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| Format: | Preprint |
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2019
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| _version_ | 1866911404606357504 |
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| author | Doan, Aleksander Walpuski, Thomas |
| author_facet | Doan, Aleksander Walpuski, Thomas |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1910_12338 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | Counting embedded curves in symplectic 6-manifolds Doan, Aleksander Walpuski, Thomas Symplectic Geometry 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. |
| title | Counting embedded curves in symplectic 6-manifolds |
| topic | Symplectic Geometry |
| url | https://arxiv.org/abs/1910.12338 |