Duality for $p$-adic geometric pro-\'etale cohomology

Abstract

We prove that $p$-adic geometric pro-\'etale cohomology of smooth partially proper rigid analytic varieties over $p$-adic fields seen in the category of Topological Vector Spaces satisfies a Poincar\'e duality as we have conjectured. This duality descends, via fully-faithfulness results of Colmez-Nizio{\l}, from a Poincar\'e duality for solid quasi-coherent sheaves on the Fargues-Fontaine curve representing this cohomology. The latter duality is proved by passing, via comparison theorems, to analogous sheaves representing syntomic cohomology and then reducing to Poincar\'e duality for ${\mathbf B}^+_{\rm st}$-twisted Hyodo-Kato and filtered $\mathbf{B}^+_{\rm dr}$-cohomologies that, in turn, reduce to Serre duality for smooth Stein varieties -- a classical result.