For a closed oriented smooth
–manifold
with
,
the Seiberg–Witten invariants are well-defined. Taubes’
“” theorem
asserts that if
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
–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
–forms,
and it is shown that they recover the Seiberg–Witten invariants over
. This is an extension
of “” to nonsymplectic
–manifolds.
The main result of this paper asserts the following. Given a suitable near-symplectic form
and tubular
neighborhood
of its zero set, there are well-defined counts of pseudoholomorphic curves in a completion of the
symplectic cobordism
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
.