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.
- http://www.mathematik.hu-berlin.de/~berg/Almost_Complex_Manifolds_Seminar_2011_03_07.pdf ↩
- http://en.wikipedia.org/wiki/Talk:Almost_complex_manifold#Holomorphic.3F ↩
- 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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