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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