More constraints on Symplectic forms from Seiberg-Witten invariants

Abstract

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 which reduces mod(2) to the 2nd Steiffel-Whitney class of the mani-fold. A brief mathematical description of these new invariants is given in the recent preprint [KM1]. Using the Seiberg-Witten equations, I proved in [T] the following: Theorem 1. Let X be a compact, oriented, 4 dimensional manifold with b 2+ ≥ 2. Let ω be a symplectic form on X with ω ∧ω giving the orientation. Then the first Chern class of the canonical bundle of a compatible, almost complex structure on X has Seiberg-Witten invariant equal to ±1. (A corollary of this theorem is the assertion that connect sums of non-negative definite compact, oriented 4-manifolds do not admit symplectic forms which are compatible with the orientation.) Subsequently, I have found that a slight modification of the proof of Theorem 1 gives further results about symplectic 4-manifolds. The purpose of this note is to report on these additional results. The first result below constrains the other cohomology classes on X which have non-zero Seiberg-Witten invariant. In the theorem below, [ω] denotes the cohomology class of the symplectic form ω, and K → X is the canonical bundle for any almost complex structure on X which is compatible with ω. Also, the symbol • denotes the bilinear pairing on cohomology as given by cup product and evaluation on the fundamental class of X. Theorem 2. Let X be a compact, oriented symplectic manifold with b 2+ ≥ 2 and with symplectic form ω which is compatible with the given orientation. Let c ∈ H 2 (X; Z) have non-zero Seiberg-Witten invariant. Then