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If M is a complex manifold with complex structure J, why the cotangent bundle of M carries 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
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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 Lemma 4.1 of Lee's Introduction to Smooth Manifolds 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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