Gromov-Witten theory of bicyclic pairs
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
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2023
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| _version_ | 1866908432636837888 |
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| author | van Garrel, Michel Nabijou, Navid Schuler, Yannik |
| author_facet | van Garrel, Michel Nabijou, Navid Schuler, Yannik |
| contents | A bicyclic pair is a smooth surface equipped with a pair of smooth divisors intersecting in two reduced points. Resolutions of self-nodal curves constitute an important special case. We investigate the logarithmic Gromov-Witten theory of bicyclic pairs. We establish correspondences with local Gromov-Witten theory and open Gromov-Witten theory in all genera, a correspondence with orbifold Gromov-Witten theory in genus zero, and correspondences between all-genus refined Gopakumar-Vafa invariants and refined quiver Donaldson-Thomas invariants. For self-nodal curves in $\mathbb{P}(1,1,r)$ we obtain closed formulae for the genus zero invariants and relate these to the invariants of local curves. We also establish a conceptual relationship between invariants relative a self-nodal plane cubic and invariants relative a smooth plane cubic. The technical heart of the paper is a qualitatively new analysis of the degeneration formula for stable logarithmic maps, involving a tight intertwining of tropical and intersection-theoretic vanishing arguments. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_06058 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Gromov-Witten theory of bicyclic pairs van Garrel, Michel Nabijou, Navid Schuler, Yannik Algebraic Geometry 14N35 (Primary), 14H10, 14N10, 14A21 (Secondary) A bicyclic pair is a smooth surface equipped with a pair of smooth divisors intersecting in two reduced points. Resolutions of self-nodal curves constitute an important special case. We investigate the logarithmic Gromov-Witten theory of bicyclic pairs. We establish correspondences with local Gromov-Witten theory and open Gromov-Witten theory in all genera, a correspondence with orbifold Gromov-Witten theory in genus zero, and correspondences between all-genus refined Gopakumar-Vafa invariants and refined quiver Donaldson-Thomas invariants. For self-nodal curves in $\mathbb{P}(1,1,r)$ we obtain closed formulae for the genus zero invariants and relate these to the invariants of local curves. We also establish a conceptual relationship between invariants relative a self-nodal plane cubic and invariants relative a smooth plane cubic. The technical heart of the paper is a qualitatively new analysis of the degeneration formula for stable logarithmic maps, involving a tight intertwining of tropical and intersection-theoretic vanishing arguments. |
| title | Gromov-Witten theory of bicyclic pairs |
| topic | Algebraic Geometry 14N35 (Primary), 14H10, 14N10, 14A21 (Secondary) |
| url | https://arxiv.org/abs/2310.06058 |