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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?2012-10-02
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    I mean viewed as a manifold.2012-10-02
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    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?2012-10-02
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    The same, but I had some question about that, I was little bit confused why that question. Thank you.2012-10-04

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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 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.