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

Abstract

For a closed oriented smooth $4$–manifold $X$ with $b_{+}^2\left(X\right)>0$, 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 $\mathbb{Z}∕2\mathbb{Z}$. 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 $\omega$ and tubular neighborhood $\mathcal{𝒩}$ of its zero set, there are well-defined counts of pseudoholomorphic curves in a completion of the symplectic cobordism $\left(X-\mathcal{𝒩},\omega \right)$ 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 $X$.