Intersection of almost complex submanifolds

We show the intersection of a compact almost complex sub- variety of dimension 4 and a compact almost complex submanifold of codimension 2 is a J -holomorphic curve. This is a generalization of positivity of intersections for J -holomorphic curves in almost complex 4-manifolds to higher dimensions. As an application, we discuss pseudo- holomorphic sections of a complex line bundle. We introduce a method to produce J-holomorphic curves using the differential geometry of almost Hermitian manifolds. When our main result is applied to pseudoholomorphic maps, we prove the singularity subset of a pseudoholomorphic map between almost complex 4-manifolds is J -holomorphic. Building on this, we show degree one pseudoholomorphic maps between almost complex 4-manifolds are actually birational morphisms in pseudoholomorphic category