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If $M$ is a complex manifold with complex structure $J$, why does the cotangent bundle of $M$ carry a natural complex structure, and not an almost complex structure. Is that obvious?

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Do you mean the cotangent bundle viewed as a vector bundle, is a complex vector bundle, or do you mean that the cotangent bundle viewed as a manifold is a complex manifold? – Michael Albanese Oct 2 '12 at 7:45
I mean viewed as a manifold. – Novak Djokovic Oct 2 '12 at 7:57
I guess it is ovious, using holomorphic charts for $M$ you can presumably define holomorphic charts for $T^*M$, the usual charts will do I believe. What have you tried? – Olivier Bégassat Oct 2 '12 at 9:16
The same, but I had some question about that, I was little bit confused why that question. Thank you. – Novak Djokovic Oct 4 '12 at 13:47
up vote 1 down vote accepted

As Olivier suggests, you can define a complex structure on $T^*M$ directly using the complex structure on $M$. This is basically the complex version of Proposition $3.18$ of Lee's Introduction to Smooth Manifolds (second edition) except that instead of $T^*M$, Lee is using $TM$. If you would like further information on this approach, let me know.

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