Doan, Aleksander;
Walpuski, Thomas;
(2023)
Counting embedded curves in symplectic 6-manifolds.
Commentarii Mathematici Helvetici
, 98
(4)
pp. 693-769.
10.4171/CMH/556.
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Abstract
Based on computations of Pandharipande (1999), Zinger (2011) proved that the Gopakumar–Vafa BPS invariants 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≫1, thus proving the Gopakumar–Vafa finiteness conjecture for primitive Calabi–Yau classes and arbitrary Fano classes.
| Type: | Article |
|---|---|
| Title: | Counting embedded curves in symplectic 6-manifolds |
| Open access status: | An open access version is available from UCL Discovery |
| DOI: | 10.4171/CMH/556 |
| Publisher version: | http://dx.doi.org/10.4171/cmh/556 |
| Language: | English |
| Additional information: | Copyright © 2023 Swiss Mathematical Society Published by EMS Press. This work is licensed under a CC BY 4.0 license, https://creativecommons.org/licenses/by/4.0/. |
| Keywords: | Pseudo-holomorphic curves; curve counting; Gopakumar–Vafa finiteness |
| UCL classification: | UCL UCL > Provost and Vice Provost Offices > UCL BEAMS UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences UCL > Provost and Vice Provost Offices > UCL BEAMS > Faculty of Maths and Physical Sciences > Dept of Mathematics |
| URI: | https://discovery-pp.ucl.ac.uk/id/eprint/10190825 |
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