The geometric Sen morphism is the unique lift of the Kodaira--Spencer morphism

TL;DR

This paper proves that the geometric Sen morphism uniquely lifts the Kodaira--Spencer morphism for de Rham torsors over p-adic varieties, linking it to the derivative of the lattice Hodge period map and extending previous minuscule case results.

Contribution

It establishes the uniqueness of the lift of the Kodaira--Spencer morphism by the geometric Sen morphism in a broad setting, including non-minuscule period domains.

Findings

The geometric Sen morphism is the unique lift of the Kodaira--Spencer morphism.

It is the derivative of the lattice Hodge period map.

Application to non-minuscule period domains generalizing local Shimura varieties.

Abstract

We show that the geometric Sen morphism of a de Rham torsor over a smooth rigid analytic variety over a $p$-adic field is the unique lift, along a natural map, of the Kodaira--Spencer morphism of the associated filtered torsor with integrable connection. This extends previous computations in the minuscule case, and implies that the geometric Sen morphism is the derivative of the lattice Hodge period map. The computation applies, in particular, to non-minuscule period domains generalizing local Shimura varieties, furnishing new examples of towers satisfying He's stalkwise perfectoidness.