Diffeomorphism that preserves almost complex structure on complex manifold

Question

I have a real newbie question here.

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Is it true that if we have a diffeomorphism of a complex manifold that preserves induced almost complex structure than it is a biholomorphism? (I assume that the inverse map also preserves almost complex structure).

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It also would be very useful if somebody could give some basic references so I can look up similar simple questions there.

Answer 1

Yes, it is true, it is a corollary of the inverse function theorem in complex analysis. If it is a diffeomorphism, locally, its complex Jacobian is invertible and you can apply:

https://en.wikipedia.org/wiki/Inverse_function_theorem#Holomorphic_functions