Intersection of almost complex submanifolds
Intersection of almost complex submanifolds
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DOI：
10.4310/cjm.2018.v6.n4.a2
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发表时间：
2017-07
期刊：
影响因子：
--
通讯作者：
Weiyi Zhang
中科院分区：
文献类型：
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作者：
Weiyi Zhang
我们证明了维度 $4$ 的紧近复子流形和余维 $2$ 的紧近复子流形的交集是 $J$ 全纯曲线。这是几乎复杂的 $4$-流形中 $J$-全纯曲线的交集正性推广到更高维度。作为一个应用，我们讨论复杂线束的伪全纯截面。我们引入了一种使用几乎埃尔米特流形的微分几何生成$J$-全纯曲线的方法。当我们的主要结果应用于伪全纯映射时，我们证明了几乎复杂的$4$-流形之间的伪全纯映射的奇点子集是$J$-全纯的。在此基础上，我们展示了几乎复杂的 $4$ 流形之间的一级伪全纯映射实际上是伪全纯范畴中的双有理态射。
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.