Effective classes and Lagrangian tori in symplectic four-manifolds

Abstract

An effective class in a closed symplectic four-manifold is a twodimensional homology class which is realized by a J-holomorphic cycle for every tamed almost complex structure J. We first prove that effective classes are orthogonal to Lagrangian tori with respect to the intersection form. We then deduce an invariant under birational transformations of closed symplectic four-manifolds. We finally prove using the same techniques of symplectic field theory that the unit cotangent bundle of a compact orientable hyperbolic Lagrangian surface does not embed as a hypersurface of contact type in a rational or ruled symplectic four-manifold.