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Constructing Parabolic Non-Abelian Hodge Correspondence in Positive Characteristic Using Parabolic Bases

Xiaojin Lin

TL;DR

The paper introduces parabolic bases to localize the study of parabolic bundles and parabolic $\lambda$-connections, enabling a parabolic non-abelian Hodge correspondence in positive characteristic and a rank-2 Higgs–de Rham flow. It constructs parabolic Cartier and inverse Cartier transforms via an exponential twist, establishing tensor equivalences between parabolic $\lambda$-connections and parabolic bundles, and proving functorial properties such as compatibility with pullback and pushforward, including BIS-type correspondences. A centerpiece is the rank-2 parabolic Higgs–de Rham flow, for which an algorithm determines the maximal destabilizing subbundle; the extension class is described by $\xi = F^{*}(\theta) \cup \kappa$, where $\kappa$ is the Deligne–Illusie class, connecting to Li–Sheng-type periodicity questions. The framework extends the BIS correspondence to parabolic and higher-dimensional contexts, clarifies the role of exponential twists, and points toward broader implications for $p$-adic Hodge theory and root-stack geometry.

Abstract

We introduce the concept of parabolic bases to establish a localized framework for parabolic bundles and parabolic $λ$-connections. Building on this foundation, we propose a novel method for constructing the parabolic non-abelian Hodge correspondence in positive characteristic, extending the work originally developed by Krishnamoorthy and Sheng for algebraic curves. Additionally, we investigate the rank $2$ parabolic Higgs-de Rham flow operator and present a modified version of the Sun-Yang-Zuo algorithm, specifically adapted to the parabolic setting.

Constructing Parabolic Non-Abelian Hodge Correspondence in Positive Characteristic Using Parabolic Bases

TL;DR

The paper introduces parabolic bases to localize the study of parabolic bundles and parabolic -connections, enabling a parabolic non-abelian Hodge correspondence in positive characteristic and a rank-2 Higgs–de Rham flow. It constructs parabolic Cartier and inverse Cartier transforms via an exponential twist, establishing tensor equivalences between parabolic -connections and parabolic bundles, and proving functorial properties such as compatibility with pullback and pushforward, including BIS-type correspondences. A centerpiece is the rank-2 parabolic Higgs–de Rham flow, for which an algorithm determines the maximal destabilizing subbundle; the extension class is described by , where is the Deligne–Illusie class, connecting to Li–Sheng-type periodicity questions. The framework extends the BIS correspondence to parabolic and higher-dimensional contexts, clarifies the role of exponential twists, and points toward broader implications for -adic Hodge theory and root-stack geometry.

Abstract

We introduce the concept of parabolic bases to establish a localized framework for parabolic bundles and parabolic -connections. Building on this foundation, we propose a novel method for constructing the parabolic non-abelian Hodge correspondence in positive characteristic, extending the work originally developed by Krishnamoorthy and Sheng for algebraic curves. Additionally, we investigate the rank parabolic Higgs-de Rham flow operator and present a modified version of the Sun-Yang-Zuo algorithm, specifically adapted to the parabolic setting.
Paper Structure (23 sections, 26 theorems, 154 equations, 3 tables)

This paper contains 23 sections, 26 theorems, 154 equations, 3 tables.

Table of Contents

  1. Introduction
  2. Motivation and Background
  3. Main Results
  4. Organization
  5. Local Theory for Parabolic Bundles
  6. Vector Bundles with Parabolic bases
  7. The correspondence between the category PVB and the category of parabolic bundles
  8. Local theory of parabolic lambda-connections
  9. Parabolic lambda-connections
  10. Pullback and Pushforward of Parabolic lambda-connections
  11. Pullback of parabolic lambda-connection
  12. Pushforward of parabolic lambda-connection
  13. Parabolic Cartier and Parabolic Inverse Cartier transformation
  14. Revisit of exponential twist
  15. Inverse Cartier Transformation via Exponential Twist
  16. ...and 8 more sections

Key Result

Theorem 1.1

The category of vector bundles with parabolic bases over $(X,D)$ is tensor equivalent to the category of parabolic bundles over $(X,D)$.

Theorems & Definitions (71)

  • Theorem 1.1: =Theorem \ref{['thm of equivalence between two category']}
  • Theorem 1.2: =Theorem \ref{['thm of equiv between adjusted par and nil conn with par bas']}
  • Theorem 1.3: =Theorem \ref{['thm of Bis for vb with pb']}
  • Theorem 1.4: =Theorem \ref{['thm of Cartier Descent']}
  • Theorem 1.5: =Theorem \ref{['thm of catier correspondence']}
  • Theorem 1.6: =Theorem \ref{['thm of par inverse commutes with pullback and push forward']}
  • Theorem 1.7: =Theorem \ref{['thm of ext class']}
  • Definition 2.1
  • Definition 2.2: Local model
  • Definition 2.3
  • ...and 61 more