On the cohomology of p-adic analytic spaces, II: The $C_{\rm st}$-conjecture

Commets are welcome !Fontaine has formulated conjectures (which are now theorems) relating \'etale and de Rham cohomologies of algebraic varieties over a $p$-adic field $K$. We state analogs of Fontaine's conjectures for smooth dagger varieties over $K$ or the completion $C$ of its algebraic closure and we prove these conjectures for varieties which are well-behaved at infinity (which includes quasi-compact varieties, analytifications of algebraic varieties, Stein varieties or, more generally, some holomorphically convex varieties). For varieties over $K$, the ``pro-\'etale-to-de Rham'' direction of these conjectures explains how to recover de Rham cohomology from pro-\'etale cohomology using the action of the absolute Galois group of $K$ (the proof uses, among other tools, Fontaine's theory of almost $C$-representations). For varieties over $C$, this Galois action is replaced by a ``geometrization'' of pro-\'etale cohomology (viewed as the $C$-points of a pro-\'etale sheaf on the category of perfectoid spaces over $C$). This geometrization plays a large role in the proof of the ``de Rham-to-pro-\'etale'' direction of the conjectures and makes it possible to appeal to the theory of Banach-Colmez spaces as a substitute for Fontaine's almost $C$-representations