SW ⇒ Gr: FROM THE SEIBERG-WITTEN EQUATIONS TO PSEUDO-HOLOMORPHIC CURVES

The purpose of this article is to explain how pseudo-holomorphic curves in a symplectic 4-manifold can be constructed from solutions to the Seiberg-Witten equations. As such, the main theorem proved here (Theorem 1.3) is an existence theorem for pseudo-holomorphic curves. This article thus provides a proof of roughly half of the main theorem in the announcement [T1]. That theorem, Theorem 4.1, asserts an equivalence between the Seiberg-Witten invariants for a symplectic manifold and a certain Gromov invariant which counts (with signs) the number of pseudoholomorphic curves in a given homology class. The Seiberg-Witten invariants were introduced to mathematicians by Witten [W] based on his joint work with Nat Seiberg [SW1], [SW2]. A description of these invariants is given in Section 1. (See also [KM1], [T1].) Suffice it to say here that when X is a compact, oriented, 4-dimensional manifold with