Uniqueness of symplectic canonical class, surface cone and symplectic cone of 4-manifolds with \(B^+=1\) (Q1424016)

The paper studies canonical classes and symplectic classes. Among the main results, the authors prove that if \(M\) is a smooth minimal oriented 4-manifold with \(b^+=1\) and admitting symplectic structures, then the (symplectic) canonical classes are unique up to sign and any real degree-2 cohomology classes of positive squares are symplectic classes, namely they can be represented by symplectic forms. There are also results for non-minimal manifolds. Previously, C. McMullen and C. Taubes, I. Smith, S. Vidussi and others have shown that for the \(b^+>1\) case, the canonical classes are in general not unique on smooth 4-manifolds which admit symplectic structures (although they are if admitting Kähler structures). The main tools used in the paper include Seiberg-Witten invariants, Gromov-Taubes invariants and their relations.