On Donaldson's 4-6 question
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2023
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| _version_ | 1866913787735441408 |
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| author | Hirschi, Amanda Wang, Luya |
| author_facet | Hirschi, Amanda Wang, Luya |
| contents | We prove that the examples by Smith and McMullen-Taubes provide infinitely many counterexamples to one direction of Donaldson's 4-6 question and the closely related Stabilising Conjecture. These are the first known counterexamples. In the other direction, we show that the Gromov-Witten invariants of two simply-connected closed symplectic $4$-manifolds, whose products with $(S^2,ω_{\text{std}})$ are deformation equivalent, agree. In particular, when $b_2^+ \geq 2$, these $4$-manifolds have the same Seiberg-Witten invariants. Furthermore, one can replace $(S^2,ω_{\text{std}})$ by $(S^2,ω_{\text{std}})^k$ for any $k \geq 1$ in both results. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_07041 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On Donaldson's 4-6 question Hirschi, Amanda Wang, Luya Symplectic Geometry Geometric Topology 53D05, 53D45 We prove that the examples by Smith and McMullen-Taubes provide infinitely many counterexamples to one direction of Donaldson's 4-6 question and the closely related Stabilising Conjecture. These are the first known counterexamples. In the other direction, we show that the Gromov-Witten invariants of two simply-connected closed symplectic $4$-manifolds, whose products with $(S^2,ω_{\text{std}})$ are deformation equivalent, agree. In particular, when $b_2^+ \geq 2$, these $4$-manifolds have the same Seiberg-Witten invariants. Furthermore, one can replace $(S^2,ω_{\text{std}})$ by $(S^2,ω_{\text{std}})^k$ for any $k \geq 1$ in both results. |
| title | On Donaldson's 4-6 question |
| topic | Symplectic Geometry Geometric Topology 53D05, 53D45 |
| url | https://arxiv.org/abs/2309.07041 |