Locally analytic vector bundles on the Fargues–Fontaine curve

Locally analytic vector bundles on the Fargues-Fontaine curve Gal PoratWe develop a version of Sen theory for equivariant vector bundles on the Fargues-Fontaine curve.We show that every equivariant vector bundle canonically descends to a locally analytic vector bundle.A comparison with the theory of (ϕ, )-modules in the cyclotomic case then recovers the Cherbonnier-Colmez decompletion theorem.Next, we focus on the subcategory of de Rham locally analytic vector bundles.Using the p-adic monodromy theorem, we show that each locally analytic vector bundle E has a canonical differential equation for which the space of solutions has full rank.As a consequence, E and its sheaf of solutions Sol(E) are in a natural correspondence, which gives a geometric interpretation of a result of Berger on (ϕ, )-modules.In particular, if V is a de Rham Galois representation, its associated filtered (ϕ, N , G K )-module is realized as the space of global solutions to the differential equation.A key to our approach is a vanishing result for the higher locally analytic vectors of representations satisfying the Tate-Sen formalism, which is also of independent interest.1. Introduction 899 2. Locally analytic and pro-analytic vectors 905 3. Equivariant vector bundles 909 4. Locally analytic vector bundles 911 5. Acyclicity of locally analytic vectors for semilinear representations 916 6. Descent to locally analytic vectors 931 7. The comparison with (ϕ, )-modules 935 8. Locally analytic vector bundles and p-adic differential equations 938