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For a closed oriented smooth 4-manifold X with $b^2_+(X)>0$, theSeiberg-Witten invariants are well-defined. Taubes' "SW=Gr" theorem assertsthat if X carries a symplectic form then these invariants are equal towell-defined counts of pseudoholomorphic curves, Taubes' Gromov invariants. Inthe absence of a symplectic form there are still nontrivial closed self-dual2-forms which vanish along a disjoint union of circles and are symplecticelsewhere. This paper and its sequel describes well-defined counts ofpseudoholomorphic curves in the complement of the zero-set of such"near-symplectic" forms, and it is shown that they recover the Seiberg-Witteninvariants (mod 2). This is an extension of Taubes' "SW=Gr" theorem tonon-symplectic 4-manifolds. The main result of this paper asserts thefollowing. Given a suitable near-symplectic form w, a tubular neighborhood N ofits 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 thesymplectic 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 afunction of spin-c structures on X.

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