Compact Riemann Surfaces: a Threefold Categorical Equivalence

We define and prove basic properties of Riemann surfaces, which we follow with a discussion of divisors and an elementary proof of the RiemannRoch theorem for compact Riemann surfaces. The Riemann-Roch theorem is used to prove the existence of a holomorphic embedding from any compact Riemann surface into n-dimensional complex projective space Pn. Using comparison principles such as Chow’s theorem we construct functors from the category of compact Riemann surfaces with nonconstant holomorphic maps to the category of smooth projective algebraic curves with regular algebraic maps and the category of function fields over C of transcendence degree one with morphisms of complex algebras. We conclude by proving that these functors establish a threefold equivalence of categories.