Abelian monopoles: the case of a positive-dimensional moduli space

In this paper we consider (in the framework of the general Seiberg–Witten theory) the case when the moduli space of solutions of the Seiberg–Witten equations has positive even dimension. We describe a connection between the Seiberg–Witten invariants of a given manifold $X$ and those of the connected sum $Y=X \# d\overline{\mathbb{CP}}^2$ where $d=(1/2)\operatorname{v.dim}\mathcal M_{SW}$. We introduce the notion of a complex structure with degeneration (based on the connection between spinor geometry and complex geometry) and generalize the notion of a pseudoholomorphic curve to the case when the underlying manifold a priori has no almost complex structure.