Volume 24, issue 4 (2020)

Download this article
Download this article For screen
For printing
Recent Issues

Volume 28
Issue 5, 1995–2482
Issue 4, 1501–1993
Issue 3, 1005–1499
Issue 2, 497–1003
Issue 1, 1–496

Volume 27, 9 issues

Volume 26, 8 issues

Volume 25, 7 issues

Volume 24, 7 issues

Volume 23, 7 issues

Volume 22, 7 issues

Volume 21, 6 issues

Volume 20, 6 issues

Volume 19, 6 issues

Volume 18, 5 issues

Volume 17, 5 issues

Volume 16, 4 issues

Volume 15, 4 issues

Volume 14, 5 issues

Volume 13, 5 issues

Volume 12, 5 issues

Volume 11, 4 issues

Volume 10, 4 issues

Volume 9, 4 issues

Volume 8, 3 issues

Volume 7, 2 issues

Volume 6, 2 issues

Volume 5, 2 issues

Volume 4, 1 issue

Volume 3, 1 issue

Volume 2, 1 issue

Volume 1, 1 issue

The Journal
About the Journal
Editorial Board
Editorial Procedure
Subscriptions
 
Submission Guidelines
Submission Page
Policies for Authors
Ethics Statement
 
ISSN 1364-0380 (online)
ISSN 1465-3060 (print)
Author Index
To Appear
 
Other MSP Journals
Taming the pseudoholomorphic beasts in $\mathbb{R} \times (S^1 \times S^2)$

Chris Gerig

Geometry & Topology 24 (2020) 1791–1839
Abstract

For a closed oriented smooth 4–manifold X with b+2(X) > 0, the Seiberg–Witten invariants are well-defined. Taubes’ “ SW = Gr” theorem asserts that if X carries a symplectic form then these invariants are equal to well-defined counts of pseudoholomorphic curves, Taubes’ Gromov invariants. In the absence of a symplectic form, there are still nontrivial closed self-dual 2–forms which vanish along a disjoint union of circles and are symplectic elsewhere. This paper and its sequel describe well-defined integral counts of pseudoholomorphic curves in the complement of the zero set of such near-symplectic 2–forms, and it is shown that they recover the Seiberg–Witten invariants over 2. This is an extension of “ SW = Gr” to nonsymplectic 4–manifolds.

The main result of this paper asserts the following. Given a suitable near-symplectic form ω and tubular neighborhood 𝒩 of its zero set, there are well-defined counts of pseudoholomorphic curves in a completion of the symplectic cobordism (X 𝒩,ω) which are asymptotic to certain Reeb orbits on the ends. They can be packaged together to form “near-symplectic” Gromov invariants as a function of spin-c structures on X.

Keywords
near-symplectic, Seiberg–Witten, Gromov, pseudoholomorphic, ECH
Mathematical Subject Classification 2010
Primary: 53D42
Secondary: 57R57
References
Publication
Received: 16 October 2018
Revised: 14 August 2019
Accepted: 9 November 2019
Published: 10 November 2020
Proposed: Simon Donaldson
Seconded: Ciprian Manolescu, Leonid Polterovich
Authors
Chris Gerig
Mathematics Department
University of California, Berkeley
Berkeley, CA
United States