Perfectoidness criterion via tangent spaces for infinite-level torsors

Establish that for a K-torsor \tilde{Z} over a smooth rigid analytic variety Z/L (as introduced in the paper), the base change \tilde{Z}_C to C=\overline{L}^{\wedge} is perfectoid whenever, for every geometric point \tilde{z}, the inscribed tangent space T_{\tilde{z}\tilde{Z}}=\overline{T_{\tilde{z}\tilde{Z}^{\,}}} is a perfectoid Banach–Colmez space. In other words, determine whether perfectoidness of \tilde{Z}_C is implied by perfectoidness of all inscribed tangent spaces at geometric points.

Background

The paper constructs inscribed structures and tangent bundles for K-torsors \tilde{Z} over smooth rigid analytic varieties Z/L and analyzes their derivatives and period maps. In this setting, the inscribed tangent spaces at geometric points are Banach–Colmez spaces, for which a classification of perfectoid objects is available.

Motivated by this differential framework and by recent progress on geometric Sen theory, the authors formulate a criterion conjecturing that global perfectoidness of the infinite-level torsor \tilde{Z}_C is controlled by local perfectoidness of its inscribed tangent spaces. They note this criterion aligns with a conjecture of Rodriguez Camargo expressed in Sen-theoretic terms.