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Degree two Gopakumar–Vafa invariants of local curves

Ben Davison and Naoki Koseki
Geometry & Topology 30 (2026) 307–336
DOI: 10.2140/gt.2026.30.307
Abstract

We investigate the Gopakumar–Vafa (GV) theory of local curves, namely, the total spaces of rank two vector bundles with canonical determinant on smooth projective curves. Under a certain genericity condition on the rank two bundles, we propose a general mechanism to compute the degree two GV invariants of local curves. In particular, we determine all the degree two GV invariants when the base curve has genus two. Combined with previous work by Bryan and Pandharipande, we obtain the GV/GW correspondence in this case. When the base curve has genus greater than two, we calculate GV invariants for some extremal genera, providing evidence for the GV/GW conjecture for curves of higher genus.

Keywords
Gopakumar–Vafa theory, Gromov–Witten theory, perverse sheaves
Mathematical Subject Classification
Primary: 14J32, 14N35
Secondary: 14F06, 14M12
References
Publication
Received: 23 September 2024
Revised: 19 April 2025
Accepted: 7 June 2025
Published: 19 January 2026
Proposed: Jim Bryan
Seconded: Mark Gross, Richard P Thomas
Authors
Ben Davison
School of Mathematics and Maxwell Institute for Mathematical Sciences
University of Edinburgh
Edinburgh
United Kingdom
Naoki Koseki
The University of Liverpool
Liverpool
United Kingdom
© 2026 MSP (Mathematical Sciences Publishers).

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