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I'm looking for a reference that proves implicit function theorem for polynomials in two variables over the complex numbers via the real version. Such a theorem is needed, for example, in the theory of algebraic curves, in order to construct charts to prove they form a complex manifold.

Also apparently a higher dimension version is useful for dealing with complete intersection curves in $\mathbb{P}^n$. I would also appreciate any reference on that.

I've seen proofs of the implicit function theorem for real spaces, for example in Spivak's Calculus on Manifolds, but I've never been able to find a proof of the complex version.

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up vote 4 down vote accepted

You can find the proof in the book Complex Geometry: An Introduction by Daniel Huybrechts.

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Thanks! Just what I was looking for. – Potato Feb 24 '12 at 6:05

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