The local behaviour of holomorphic curves in almost complex 4-manifolds

In this paper we prove various results about the positivity of intersections of holomorphic curves in almost complex 4-manifolds which were stated by Gromov. We also show that the virtual genus of any closed holomorphic curve in an almost complex 4-manifold is nonnegative. These technical results form the basis of the classification of rational and ruled symplectic 4-manifolds given in