The Seiberg-Witten invariants and symplectic forms
Mathematical Research Letters
Recently, Seiberg and Witten (see [SW1] , [SW2] , [W]) introduced a remarkable new equation which gives differential-topological invariants for a compact, oriented 4-manifold with a distinguished integral cohomology class. A brief mathematical description of these new invariants is given in the recent preprint [KM]. My purpose here is to prove the following theorem: Let ω be a symplectic form on X with ω ∧ ω giving the orientation. Then the first Chern class of the associated almost complex
... almost complex structure on X has Seiberg-Witten invariant equal to ±1. (Note: There are no symplectic forms on X unless b 2+ and the first Betti number of X have opposite parity.) In a subsequent article with joint authors, a vanishing theorem will be proved for the Seiberg-Witten invariants of a manifold X, as in the theorem, which can be split by an embedded 3-sphere as X − ∪ X + where neither X − nor X + have negative definite intersection forms. Thus, no such manifold admits a symplectic form. That is, Corollary. Connect sums of 4-manifolds with non-negative definite intersection forms do not admit symplectic forms which are compatible with the given orientation. For example, when n > 1 and m ≥ 0, then ( #n CP 2 )#( #m CP 2 ) has no symplectic form which defines the given orientation. The Main Theorem also implies that the Seiberg-Witten invariant for the canonical class of a complex surface with b 2+ ≥ 3 is equal to ±1. However, this result is easy to prove directly, as there is just one nondegenerate solution in this case.