The Riemann existence theorem

Finite étale covers of a scheme of finite type over the complex numbers are the same as finite covering spaces of its complex points, with no normality hypothesis: covers of punctured polydiscs, the smooth case through compactification and GAGA, descent along proper surjective morphisms, the general theorem by descent along the resolution, its analytic form, and the comparison of étale and topological fundamental groups.

Six lessons with proofs, examples and exercises with solutions. Written and self-checked by Claude Opus 5.5 (Anthropic); independent review is not complete.

1. Finite étale covers and their analytification
2. Connectedness and full faithfulness
3. Finite covers of punctured polydiscs
4. Covers of smooth varieties
5. Descent along proper surjective morphisms
6. The Riemann existence theorem

Theorems and assumptions · Reader and editable sources · Course record

Prerequisites

• Complex analytic spaces and analytification · Cohomology of quasi-coherent sheaves
• Serre's comparison theorems and Chow's theorem · Cohomology of quasi-coherent sheaves
• Holomorphic functions of several variables
• Principalization and resolution
• Smooth blow-ups and transforms of ideals
• Facts on finite étale morphisms, henselian rings, descent and Galois categories are cited by tag from the Stacks project, through the AI-integrated edition of the Stacks project.
• Covering spaces and fundamental groups: the core course Algebraic Topology.
• Relative subanalytic triangulation · contractible neighbourhoods at singular points