Intersection of almost complex submanifolds

2017/07/26 by Zhang, Weiyi · 1 citation
#Complex Variables (math.CV) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Topology (math.GT) #Symplectic Geometry (math.SG)

paper · doi:10.48550/arxiv.1707.08253

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.

Cited by

• Kodaira dimension of almost complex 4-manifolds with torsion first Chern class

Related

• Positivity of Intersections and Tameness of Almost Complex 4-manifolds
• From Smooth to Almost Complex
• J-holomorphic curves from closed J-anti-invariant forms
• Embedded surfaces and almost complex structures
• Almost holomorphic curves in real analytic hypersurfaces