Lefschetz pencils and the canonical class for symplectic four-manifolds

We present a new proof of a result due to Taubes: if (X,ω) is a closed symplectic four-manifold with b+(X)>1+b1(X) and λ[ω] ∈ H2(X;Q) for some λ ∈ R+, then the Poincaré dual of KX may be represented by an embedded symplectic submanifold. The result builds on the existence of Lefschetz pencils on symplectic four-manifolds. We approach the topological problem of constructing submanifolds with locally positive intersections via almost-complex geometry. The crux of the argument is that a Gromov invariant counting pseudoholomorphic sections of an associated bundle of symmetric products is non-zero.