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arXiv · 2309.07041

On Donaldson's 4-6 question

Amanda Hirschi·Luya Wang
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Abstract

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 $(S^2,\omega_{\text{std}})$ are deformation equivalent, agree. In particular, when $b_2^+ \geq 2$, these $4$-manifolds have the same Seiberg-Witten invariants. Furthermore, one can replace $(S^2,\omega_{\text{std}})$ by $(S^2,\omega_{\text{std}})^k$ for any $k \geq 1$ in both results.

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Amanda Hirschi, Luya Wang. 2023-09-13. On Donaldson's 4-6 question. https://doi.org/10.2140/gt.2026.30.3145

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