ABSTRACT

Our goal in this chapter is to construct Galois extensions of C(T ), using the tools of topology and analytic geometry. To carry out this program, we need a bridge between analysis and algebra.

Theorem 6.1.1 (GAGA principle) Let X, Y be projective algebraic varieties over C, and let Xan, Y an be the corresponding complex analytic spaces. Then 1. Every analytic map Xan −→ Y an is algebraic. 2. Every coherent analytic sheaf over Xan is algebraic, and its algebraic cohomology coincides with its analytic one.