Intersection of almost complex submanifolds

Weiyi Zhang University of Warwick

Complex Variables and Complex Analysis Differential Geometry Geometric Analysis and Geometric Topology mathscidoc:2005.08001

Cambridge Journal of Mathematics, 6, (4), 451-496, 2018
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.
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@inproceedings{weiyi2018intersection,
  title={Intersection of almost complex submanifolds},
  author={Weiyi Zhang},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20200530210401865517695},
  booktitle={Cambridge Journal of Mathematics},
  volume={6},
  number={4},
  pages={451-496},
  year={2018},
}
Weiyi Zhang. Intersection of almost complex submanifolds. 2018. Vol. 6. In Cambridge Journal of Mathematics. pp.451-496. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20200530210401865517695.
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