# Connection between algebraic geometry and complex analysis?

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

• 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 – user64540 Mar 13 '15 at 14:25

## 2 Answers

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.

• you cite incorrectly; the correct citation: "In effect it says precisely which complex manifolds are defined by homogeneous polynomials." – user64540 Mar 13 '15 at 15:56
• 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) – shen Mar 13 '15 at 17:48
• 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? – Dario Mar 13 '15 at 23:18

May I recommend Mme. Raynaud: Géométrie Algébrique et Géometrie Analytique, Exposé XII. The GAGA-relation is made precise by a functor

$$an: \underline{Al}_\mathbb{C} \xrightarrow{} \underline{An},\ X \mapsto X^{an},$$

from the category of schemes over $$\mathbb{C}$$ which are locally of finite type and the category of complex spaces. $$X^{an}$$ can be considered as $$X(\mathbb{C})$$, the set of complex points of $$X$$ provided with a canonical complex structure: Raynaud shows that $$X^{an}$$ solves a universal problem, i.e. represents a certain functor. Hence $$X^{an}$$ is uniquely determined which facilitates many GAGA-constructions.

GAGA prompts the question which complex spaces $$Y$$ have the form $$Y=X^{an}$$, i.e. can be investigated by methods from algebraic geometry. Here Kodairas embedding theorem for compact manifolds $$Y$$ with a positive line bundle was the first result, a far reaching generalization of the embedding theorem for compact Riemann surfaces. Kodaira's theorem has been generalized by Grauert to compact complex spaces.

Raynaud's paper provides severals lists of properties of schemes and morphisms which carry over by the functor $$an$$. The paper is well written.