Riemann Surfaces

We begin this chapter with the definition of a Riemann surface. Let B be a compact connected Riemann surface, and $${{\mathscr {M}}}(B)$$ the field of meromorphic functions on B. If X is a compact connected Riemann surface over B (i.e. equipped with a non constant morphism $$\pi :X\rightarrow B$$ ), then the field $${{\mathscr {M}}}(X)$$ is a finite extension of $${{\mathscr {M}}}(B)$$ . Moreover, there is a finite subset $$\Delta $$ of B such that $$X'=X- \pi ^{-1}(\Delta )$$ is a connected finite cover of $$B-\Delta $$ . The functors $$X\mapsto {{\mathscr {M}}}(X)$$ and $$X\mapsto X'$$ give an equivalence between the category $${{\mathscr {V}}}^1_B$$ $${{\mathscr {V}}}^1_B$$ of compact connected Riemann surfaces over B, and respectively, the opposite category of the category of finite extensions of $${{\mathscr {M}}}(B)$$ , and the direct limit category of categories of connected covers of $$B-\Delta $$ , where $$\Delta \subset B$$ is finite.