殆复流形上可以定义全纯向量丛么

Problem . 我们知道, 在复流形$M$上, 可以证明$T^{1,0}M$是$M$上的一个全纯向量丛.

那么对一般的殆复流形, 我们是否还有这个结论成立呢?

首先, 也许要看看殆复流形上能不能定义全纯映射. 这是可以的, 用外微分算子限制到$T^{1,0}M$部分即可, 看起来我们似乎可以定义殆复流形上的全纯向量丛. 但是查阅文献却基本是否定的答案.

在文1中,

Note, that $X$ is in general only a differentiable manifold and thus there is
no concept of holomorphy: The holomorphic tangent bundle $T^{1,0} X$ can not be
a holomorphic vector bundle on a differentiable manifold. If $X$ is complex,
however, one has that $T^{1,0} X = TX$ is indeed the holomorphic tangent bundle
of $X$.

在wiki中有个Talk2, 提到locally holomorphic vector bundle可以定义.

也许文章Nonorientable manifolds, complex structures, and holomorphic vector bundles3值得一看.


  1. http://www.mathematik.hu-berlin.de/~berg/Almost_Complex_Manifolds_Seminar_2011_03_07.pdf
  2. http://en.wikipedia.org/wiki/Talk:Almost_complex_manifold#Holomorphic.3F
  3. Biswas, Indranil, and Avijit Mukherjee. “Nonorientable manifolds, complex structures, and holomorphic vector bundles.” Acta Applicandae Mathematica 69.1 (2001): 25-42.

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