  We construct parabolic analogues of (global) eigenvarieties, of patched
eigenvarieties and of (local) trianguline varieties, that we call respectively
Bernstein eigenvarieties, patched Bernstein eigenvarieties, and Bernstein
paraboline varieties. We study the geometry of these rigid analytic spaces, in
particular (generalizing results of Breuil-Hellmann-Schraen) we show that their
local geometry can be described by certain algebraic schemes related to the
generalized Grothendieck-Springer resolution. We deduce several local-global
compatibility results, including a classicality result (with no trianguline
assumption at $p$), and new cases towards the locally analytic socle conjecture
of Breuil in the non-trianguline case.
