Seiberg–Witten and Gromov invariants for self-dual harmonic 2–forms

Gerig, Chris

doi:10.2140/gt.2022.26.3307

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Seiberg–Witten and Gromov invariants for self-dual harmonic $2$–forms

Chris Gerig

Geometry & Topology 26 (2022) 3307–3365

DOI: 10.2140/gt.2022.26.3307

Abstract

Our earlier work (Geom. Topol. 24 (2020) 1791–1839) gives an extension of Taubes’ “ $SW = Gr$ ” theorem to nonsymplectic $4$ –manifolds. The main result of this sequel asserts the following: whenever the Seiberg–Witten invariants are defined over a closed minimal $4$ –manifold $X$ , they are equivalent modulo 2 to “near-symplectic” Gromov invariants in the presence of certain self-dual harmonic $2$ –forms on $X$ . A version for nonminimal $4$ –manifolds is also proved. A corollary to Morse theory on $3$ –manifolds is also announced, recovering a result of Hutchings, Lee, and Turaev about the $3$ –dimensional Seiberg–Witten invariants.

Keywords

near-symplectic, Seiberg–Witten, Gromov, pseudoholomorphic, ECH, monopole Floer

Mathematical Subject Classification 2010

Primary: 53D42, 57R57

References

Publication

Received: 16 October 2018

Revised: 18 April 2021

Accepted: 29 July 2021

Published: 16 March 2023

Proposed: Simon Donaldson

Seconded: Ciprian Manolescu, Leonid Polterovich

Authors

Chris Gerig
Department of Mathematics UC Berkeley Berkeley, CA United States

Volume 26, issue 8 (2022)