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Nilpotent orbit theorem in $p$-adic Hodge theory

Mohammad Reza Rahmati, Gerardo Flores

Abstract

We state and prove three orbit theorems on the period domains for the $p$-adic Hodge structure analogous to the complex case. We shall consider the variation of de Rham (resp. étale) cohomology in a family of projective varieties $f:\mathfrak{X} \to S$ defined over a p-adic field. First, we show that any nilpotent orbit in the period domain of p-adic Hodge structures converges to a semistable point (filtration) in the period domain of the p-adic Hodge structure. Furthermore, the nilpotent orbits of the limit point are asymptotic to the twisted period map [Theorem \ref{thm:nilpotent-orbit}]. The orbit theorems come with some estimates of the distance between the nilpotent orbit and the twisted period map. The distance estimate in the p-adic nilpotent orbit theorem is given concerning the non-archimedean metric and is based on the p-adic Fourier analysis of Amice-Schneider. The result is analogous to the orbit theorems of W. Schmid [\cite{Sch}-1973] on complex Hodge structures. Our proof is based on a \textit{Geometric Invariant Theory} (GIT) criterion for semi-stability (Kempf-Ness theorem) and estimates from the (Amice-Schneider) p-adic Fourier theory. We also state the $SL_2$-orbit theorem in the p-adic case, [Theorem \ref{th:homomorphism}]. Finally, we explain how the nilpotent orbit theorem should be modified and stated for a variation of the mixed Hodge structure [Theorem \ref{thm:mixed-orbit}].}

Nilpotent orbit theorem in $p$-adic Hodge theory

Abstract

We state and prove three orbit theorems on the period domains for the -adic Hodge structure analogous to the complex case. We shall consider the variation of de Rham (resp. étale) cohomology in a family of projective varieties defined over a p-adic field. First, we show that any nilpotent orbit in the period domain of p-adic Hodge structures converges to a semistable point (filtration) in the period domain of the p-adic Hodge structure. Furthermore, the nilpotent orbits of the limit point are asymptotic to the twisted period map [Theorem \ref{thm:nilpotent-orbit}]. The orbit theorems come with some estimates of the distance between the nilpotent orbit and the twisted period map. The distance estimate in the p-adic nilpotent orbit theorem is given concerning the non-archimedean metric and is based on the p-adic Fourier analysis of Amice-Schneider. The result is analogous to the orbit theorems of W. Schmid [\cite{Sch}-1973] on complex Hodge structures. Our proof is based on a \textit{Geometric Invariant Theory} (GIT) criterion for semi-stability (Kempf-Ness theorem) and estimates from the (Amice-Schneider) p-adic Fourier theory. We also state the -orbit theorem in the p-adic case, [Theorem \ref{th:homomorphism}]. Finally, we explain how the nilpotent orbit theorem should be modified and stated for a variation of the mixed Hodge structure [Theorem \ref{thm:mixed-orbit}].}

Paper Structure

This paper contains 16 sections, 11 theorems, 88 equations.

Table of Contents

  1. Introduction
  2. Contributions
  3. Tools
  4. p-adic Fourier correspondence.
  5. Mumford GIT stability
  6. Sen formalism in p-adic Hodge theory
  7. Summary
  8. Preliminaries
  9. p-adic Hodge theory
  10. Prelimineries on $p$-adic Period Domains
  11. Mumford GIT stability
  12. $p$-adic Fourier correspondence
  13. Period map
  14. Main results
  15. $p$-adic $SL_2$-orbits
  16. ...and 1 more sections

Key Result

Proposition 2.2

DOPR Let $E$ be $\phi$-isocrystals over a scheme $S$ of characteristic $p$. Then, the Newton vector of $E_s$ goes down under specialization, that is, when $E$ is of constant rank the function $s \mapsto \Vert \nu(E_s) \Vert$ is locally constant on $S$. Furthermore, for any $\nu_0$, the set $\{ s \in

Theorems & Definitions (23)

  • Example 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Example 2.4
  • Theorem 2.5
  • Remark 2.6
  • Definition 2.7
  • Theorem 2.8
  • Theorem 2.9
  • Theorem 2.10
  • ...and 13 more