Intersection of almost complex submanifolds

TL;DR

This paper generalizes the positivity of intersections for $J$-holomorphic curves in 4-manifolds to higher dimensions, showing intersections of certain almost complex subvarieties are $J$-holomorphic curves and exploring applications to pseudoholomorphic maps.

Contribution

It introduces a new method to produce $J$-holomorphic curves and extends intersection positivity to higher-dimensional almost complex manifolds.

Findings

Intersection of a 4-dimensional subvariety and a codimension 2 submanifold is a $J$-holomorphic curve.

The singularity set of a pseudoholomorphic map between 4-manifolds is $J$-holomorphic.

Degree one pseudoholomorphic maps are birational in the pseudoholomorphic category.

Abstract

We show the intersection of a compact almost complex subvariety 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 pseudoholomorphic 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.