Comparison of Hyodo-Kato and de Rham Fargues-Fontaine Cohomology Theories

Abstract: We prove that, for adic \'{e}tale motives over $\mathbb{C}p$, the vector bundles on the Fargues-Fontaine curve arising from their Hyodo-Kato cohomology coincide with their de Rham-Fargues-Fontaine cohomologies, where the latter provides an overconvergent refinement of crystalline vector bundles, albeit constructed on the generic fiber. This equivalence is established in the setting of symmetric monoidal $\infty$-categories and respects the full motivic structure. Furthermore, we enrich both realizations with Galois actions, yielding $G{\breve{\mathbb{Q}}_{p}}$-equivariant solid quasi-coherent sheaves on the Fargues-Fontaine curve; in this equivariant context, the comparison isomorphism becomes canonical. As an application, we show that the de Rham-Fargues-Fontaine cohomology of any smooth quasi-compact rigid analytic variety over $\mathbb{C}_p$ admits a finite slope-increasing filtration.