Taming the pseudoholomorphic beasts in $\mathbb{R}\times(S^1\times S^2)$ (1711.02069v3)

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.