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I've studied some complex analysis and basics in algebraic geometry (let's say over $\mathbb C $). We had been mentioned GAGA but nothing in detail. Anyway, from my beginner's point of view, I already see many parallels between both areas of study like

  • compact riemann surfaces vs. projective curves
  • regular functions being holomorphic
  • ramified coverings
  • globally holomorphic/regular maps on projective varieties/compact riemann surfaces being constant

Also, the field of elliptic functions on a torus is finitely generated, reminiscent of function fields of varieties (in fact the torus algebro-geometrically is an elliptic curve).

However, I don't see the precise correspondence. After all, the notion of morphism should be weaker than that of a holomorphic map. Or can one get a 1:1-correspondence? How would algebraic geometry help in studying all holomorphic functions and not just those taat happen to be regular. Maybe you could give me some intuition on the topic or GAGA in basic terms. Thank you

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  • $\begingroup$ you may search for "a compact complex manifold might not be a smooth projective variety", so it is not a one to one correspondence; also, singular algebraic curves are not Riemann surfaces $\endgroup$ – user64540 Mar 13 '15 at 14:25
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The Kodaira embedding theorem may be of interest. "It says (which) precisely complex manifolds are defined by homogeneous polynomials" c.f. http://en.wikipedia.org/wiki/Kodaira_embedding_theorem

Also the book Griffiths & Harris "Principles of Algebraic Geometry" does algebraic geometry from the complex viewpoint.

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    $\begingroup$ you cite incorrectly; the correct citation: "In effect it says precisely which complex manifolds are defined by homogeneous polynomials." $\endgroup$ – user64540 Mar 13 '15 at 15:56
  • $\begingroup$ I should have just copy/pasted rather than retyping! But what you have commented is actually what I intended for the answer (obviously not all complex manifolds) $\endgroup$ – shen Mar 13 '15 at 17:48
  • $\begingroup$ Thannks. More than for the spaces, i am interested in the relationship between regular and holomorphic functions. When studying for example $\mathbb P^1$, can we study all meromorphic maps as regular ones? $\endgroup$ – Dario Mar 13 '15 at 23:18

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