Étale cohomology of algebraic varieties over Stein compacta
Résumé
We prove a comparison theorem between the étale cohomology of algebraic varieties over Stein compacta and the singular cohomology of their analytifications. We deduce that the field of meromorphic functions in a neighborhood of a connected Stein compact subset of a normal complex space of dimension n has cohomological dimension n. As an application of Gal(C/R)-equivariant variants of these results, we obtain a quantitative version of Hilbert's 17th problem on compact subsets of real-analytic spaces.