On Donaldson's 4-6 question

TL;DR

This paper presents the first known counterexamples to a specific question in symplectic topology, while also establishing invariance of Gromov-Witten invariants under certain product deformations of 4-manifolds.

Contribution

It provides the first counterexamples to Donaldson's 4-6 question and the Stabilising Conjecture, and proves invariance of Gromov-Witten invariants under product deformations of symplectic 4-manifolds.

Findings

Counterexamples to Donaldson's 4-6 question and Stabilising Conjecture

Gromov-Witten invariants agree for certain deformation equivalent 4-manifolds

Invariance extends when replacing (S^2, ω_std) with its k-fold product

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.