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I have some questions about projective complex manifolds $X \subseteq \mathbb{P}_{\mathbb{C}}^n$. By applying Chow's theorem, we can consider them as projective varieties over $\mathbb{C}$. Regarding this, I have the following questions: \ \ Are the Hodge numbers of $X$, when considered as an algebraic variety, the same as when considered as a complex manifold? \ \ Is every holomorphic automorphism of $X$ as a complex manifold also an automorphism, when considered as an algebraic variety?

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Have you heard about Serre's GAGA? –  Fernando Muro Aug 30 '11 at 23:02
The answer to all of the above questions are yes by the amazing GAGA indicated in the previous comment. The reference is: Serre, Géométrie algébrique et géométrie analytique. Ann. Inst. Fourier, Grenoble (1955). If you prefer English, there is a sketch in a book by Griffiths and Adams. Also Neeman has a book which goes through it, but I haven't looked at it. –  Donu Arapura Aug 30 '11 at 23:41
The statement about automorphisms actually follows from Chow's theorem: apply it to the graph inside $X \times X$ (which is also projective). –  ulrich Aug 31 '11 at 6:16

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