Rational analytic syntomic cohomology

We define and study the rational analytic syntomification $X^{\mathrm{S}\mathrm{y}\mathrm{n}}$ of a partially proper rigid-analytic variety $X$ over ${\mathbb{Q}}_p$. We establish Poincaré duality and a theory of first Chern classes for the resulting cohomology theory, identify vector bundles on $X^{\mathrm{S}\mathrm{y}\mathrm{n}}$ with de Rham bundles on the Fargues–Fontaine curve of $X^{\mathrm{♢}}$ and recover several classical comparison theorems in $p$-adic Hodge theory. We also develop analogues of our results and constructions over ${\mathbb{C}}_p$.