Taming the pseudoholomorphic beasts in $\mathbb{R}\times(S^1\times S^2)$

TL;DR

This paper extends Taubes' SW=Gr theorem to non-symplectic 4-manifolds by defining counts of pseudoholomorphic curves in the complement of zero-sets of near-symplectic forms, linking them to Seiberg-Witten invariants.

Contribution

It introduces well-defined counts of pseudoholomorphic curves in near-symplectic 4-manifolds and shows they recover Seiberg-Witten invariants mod 2, extending Taubes' theorem beyond symplectic cases.

Findings

Defines near-symplectic Gromov invariants as functions of spin-c structures.

Establishes counts of J-holomorphic curves in the complement of near-symplectic zero-sets.

Shows these invariants recover Seiberg-Witten invariants mod 2.

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 describes well-defined counts of pseudoholomorphic curves in the complement of the zero-set of such "near-symplectic" forms, and it is shown that they recover the Seiberg-Witten invariants (mod 2). This is an extension of Taubes' "SW=Gr" theorem to non-symplectic 4-manifolds. The main result of this paper asserts the following. Given a suitable near-symplectic form w, a tubular neighborhood N of its zero-set, and a generic w-compatible almost complex structure J on X-N, there are well-defined counts of J-holomorphic curves in a completion of the symplectic cobordism (X-N, w) which are asymptotic to 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.