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On Donaldson's 4-6 question

Amanda Hirschi, Luya Wang

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 $(S^2,ω_{\text{std}})$ 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.

On Donaldson's 4-6 question

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 -manifolds, whose products with are deformation equivalent, agree. In particular, when , these -manifolds have the same Seiberg-Witten invariants. Furthermore, one can replace by for any in both results.
Paper Structure (10 sections, 24 theorems, 80 equations)

This paper contains 10 sections, 24 theorems, 80 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Background
  4. Strategy of proof.
  5. From four to higher dimensions
  6. Simply-connected examples
  7. McMullen-Taubes' example
  8. From Higher to Four Dimensions
  9. Gromov--Witten invariants
  10. Proof of Theorem \ref{['thm:same-GW']}

Key Result

Theorem 1

There exist infinitely many pairwise non-homeomorphic smooth closed simply-connected $4$-manifolds $X$ admitting symplectic forms $\omega_0$ and $\omega_1$ so that the product forms $\omega_0 \oplus \omega_{\normalfont\text{std}}^{\oplus k}$ and $\omega_1 \oplus \omega_{\normalfont\text{std}}^{\oplu

Theorems & Definitions (51)

  • Theorem 1
  • Theorem 2
  • Corollary 3
  • Corollary 4
  • Definition 1.1
  • Definition 2.1
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • ...and 41 more