Inscription, twistors, and $p$-adic periods

TL;DR

The work develops inscribed $v$-sheaves as a differentiable enhancement within $p$-adic geometry to recover internal tangent data that classical diamonds lack. It constructs inscribed analogues of Hodge, lattice Hodge, and Liu--Zhu period maps, and interprets their derivatives via the geometric Sen morphism and twistor-theoretic functors on the relative thickened Fargues--Fontaine curve. By coupling these with moduli of modifications, Hecke correspondences, and Newton strata, the authors obtain a unified, inscribed differential toolkit for infinite-level Shimura varieties, local shtukas, and related $p$-adic period domains. The framework yields explicit tangent and normal bundle calculations, links to cohomological smoothness and transversality, and a moduli-theoretic pathway to differential topology for diamonds. Altogether, this advances $p$-adic Hodge theory by threading period maps, tensor functors to twistors, and deformation-theoretic openness results through the inscribed perspective.

Abstract

We introduce the theory of inscribed $v$-sheaves, a differentiable extension of the theory of diamonds and $v$-sheaves with internal tangent bundles that are often relative inscribed Banach-Colmez spaces, then apply this theory to the study of $p$-adic periods. In particular, we construct natural inscribed versions of the Hodge and Hodge-Tate period maps and their lattice refinements for de Rham torsors, then compute the derivatives of these period maps in terms of classical structures in $p$-adic Hodge theory. These torsors include infinite level global Shimura varieties and infinite level local Shimura varieties, and for these spaces we also give another moduli-theoretic construction of the inscribed structure; the construction in the local Shimura case applies more generally to the non-minuscule moduli of mixed characterisic local shtukas with one leg. The key new ingredients in our study of inscribed structures on $p$-adic Lie group torsors over smooth rigid varieties over a $p$-adic field are the Liu-Zhu period map, a refinement of the Hodge period map whose derivative is the geometric Sen morphism/canonical Higgs field, and a closely related exact tensor functor from $\mathbb{Q}_p$-local systems to a category of twistor bundles on the relative thickened Fargues-Fontaine curve. These new structures are only visible after passing to the inscribed setting. We also discuss some possible implications of our computations in the vein of ``differential topology for diamonds."

Theorems & Definitions (215)

• Example 1.1.1
• Example 1.1.2
• Remark 1.2.1
• Remark 1.2.2
• Remark 1.2.4
• Remark 1.3.3
• Theorem 1: see \ref{['theorem.twistor-functor']}
• Remark 1.3.7
• Remark 1.3.8
• Remark 1.3.10