nLab Riemann existence theorem

Statement

For a nonsingular algebraic variety $X$ over the complex numbers, the functor $Y \mapsto Y^{an}$ which sends complex algebraic varieties to their complex analytic topology establishes an equivalence of categories between the corresponding étale sites $X_{et} \simeq X^{an}_{et}$.

Related concepts

• comparison theorem (étale cohomology)

References

• James Milne, section 21 of Lectures on Étale Cohomology

• Riemann’s existence theorem (pdf)

A related result in model theory:

• Rahim Moosa ,_A nonstandard Riemann existence theorem_, Trans. Amer. Math. Soc. 356(5):1781–1797 (2004) doi

We study elementary extensions of compact complex spaces and deduce that every complete type of dimension $1$ is internal to projective space. This amounts to a nonstandard version of the Riemann Existence Theorem, and answers a question posed by Anand Pillay.