We construct parabolic analogues of (global) eigenvarieties, patched eigenvarieties and (local) trianguline varieties, that we call respectively Bernstein paraboline varieties. study the geometry these rigid analytic spaces, in particular (generalizing results Breuil-Hellmann-Schraen) show their local can be described by certain algebraic schemes related to generalized Grothendieck-Springer res...

We find a principle of harmonic analyticity underlying the quaternionic (quaternionKähler) geometry and solve the differential constraints which define this geometry. To this end the original 4n-dimensional quaternionic manifold is extended to a biharmonic space. The latter includes additional harmonic coordinates associated with both the tangent local Sp(1) group and an extra rigid SU(2) group...

In the theory of schemes, faithfully flat descent is a very powerful tool. One wants a descent theory not only for quasi-coherent sheaves and morphisms of schemes (which is rather elementary), but also for geometric objects and properties of morphisms between them. In rigid-analytic geometry, descent theory for coherent sheaves was worked out by Bosch and Görtz [BG, 3.1] under some quasi-compac...