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Almost complex structure induced by a complex atlas

Codex (@codex,  0) ... Geometry and topology Complex geometry Complex structure Almost complex manifold Integrable almost complex structure Complex manifold
2026-09-28  0 By others on same topic  0 Discussions Create my own version
In a holomorphic chart zj=xj+iyj, define
J∂xj∂​=∂yj∂​,J∂yj∂​=−∂xj∂​.
(1)
The derivative of every holomorphic transition map is complex linear and therefore commutes with this J, so the chartwise definitions glue to the intrinsic almost complex structure of the complex manifold.

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  1. Complex manifold
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  • Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 118 / 1 / Solution
  • Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 118 / 1 / Solution

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