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Vanishing Cycles for Zariski-Constructible Sheaves on Rigid Analytic Varieties

Tong Zhou

TL;DR

This work develops a comprehensive theory of nearby and vanishing cycles for finite-coefficient Zariski-constructible sheaves on rigid analytic varieties over a non-archimedean field. It defines the cycles with monodromy actions, establishes fundamental functorial properties, and proves they preserve constructibility and admit a Milnor-fibre interpretation. Building on Beilinson's approach, it develops unipotent cycles, Beilinson gluing, and a maximal extension formalism, yielding a full perverse $t$-structure and duality results in this analytic setting. The results provide a robust framework for non-archimedean microlocal theory and open avenues for further study of gluing and duality in rigid analytic geometry.

Abstract

We develop a theory of nearby and vanishing cycles in the context of finite-coefficient Zariski-constructible sheaves over a non-archimedean field which is non-trivially valued, complete, algebraically closed, and of mixed characteristic or equal characteristic zero. Apart from basic properties, we show that they preserve Zariski-constructibility, have a Milnor fibre interpretation, satisfy Beilinson's gluing construction, are perverse t-exact, and commute with Verdier duality.

Vanishing Cycles for Zariski-Constructible Sheaves on Rigid Analytic Varieties

TL;DR

This work develops a comprehensive theory of nearby and vanishing cycles for finite-coefficient Zariski-constructible sheaves on rigid analytic varieties over a non-archimedean field. It defines the cycles with monodromy actions, establishes fundamental functorial properties, and proves they preserve constructibility and admit a Milnor-fibre interpretation. Building on Beilinson's approach, it develops unipotent cycles, Beilinson gluing, and a maximal extension formalism, yielding a full perverse -structure and duality results in this analytic setting. The results provide a robust framework for non-archimedean microlocal theory and open avenues for further study of gluing and duality in rigid analytic geometry.

Abstract

We develop a theory of nearby and vanishing cycles in the context of finite-coefficient Zariski-constructible sheaves over a non-archimedean field which is non-trivially valued, complete, algebraically closed, and of mixed characteristic or equal characteristic zero. Apart from basic properties, we show that they preserve Zariski-constructibility, have a Milnor fibre interpretation, satisfy Beilinson's gluing construction, are perverse t-exact, and commute with Verdier duality.

Paper Structure

This paper contains 5 sections, 30 theorems, 6 equations.

Table of Contents

  1. Introduction
  2. Definitions and basic properties
  3. Finiteness and a Milnor fibre interpretation
  4. Beilinson's construction and gluing
  5. Perversity and duality

Key Result

Theorem 1.1

The set-up is as above, $\Lambda=\mathbf{F}_{\ell^r}$ or $\mathbf{Z}/\ell^r$, $\ell\neq p$. Let $f: X\rightarrow\mathbf{A}^1$ be a map of rigid analytic varieties. Denote $\psi_f[-1]$ (resp. $\phi_f[-1]$) by $\Psi_f$ (resp. $\Phi_f$). Then: (1) $\Psi_f$ is perverse t-exact in the following sense: fo

Theorems & Definitions (73)

  • Theorem 1.1: Theorem \ref{['thm_psi_t_exact_duality']}, Corollary \ref{['cor_ptexactphigeneralcoefficient']}, Theorem \ref{['thm_phi_t_exact_duality']}
  • Remark 1.2
  • Definition 2.1: nearby and vanishing cycles
  • Remark 2.2
  • Remark 2.3
  • Definition 2.4: lu_duality_2019, Iwasawa twist
  • Lemma 2.5: lu_duality_2019
  • Remark 2.6
  • Lemma 2.7
  • proof
  • ...and 63 more