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Real Structures on Root Stacks and Parabolic Connections

Sujoy Chakraborty, Arjun Paul

TL;DR

This work develops a real-structure framework for root stacks and parabolic connections. It proves that a smooth complex variety X with a real structure Οƒ and a real sncd D induces a compatible real structure on the r-th root stack 𝔛 of (π’ͺ_X(D), s_D), and establishes an equivalence between real parabolic vector bundles on (X, D) and real vector bundles on (𝔛, Οƒ_𝔛), as well as between real logarithmic connections on (𝔛, 𝔇) and real parabolic connections on (X, D) with weights in $\frac{1}{r}\mathbb{Z}$; holomorphic real connections on 𝔛 correspond to real strongly parabolic connections on (X, D). The results unify real-structure considerations with Biswas-Borne correspondences and extend the parabolic-connection paradigm to real algebraic geometry, including a quotient-stack/Galois-cover perspective via Kawamata-type covers. Altogether, the paper provides a robust stack-theoretic bridge between real parabolic data and root-stack connections, enhancing tools for real nonabelian Hodge-type correspondences in higher dimensions.

Abstract

Let $D$ be a reduced effective strict normal crossing divisor on a smooth complex variety $X$, and let $\mathfrak{X}_D$ be an associated root stack over $\mathbb C$. Suppose that $X$ admits an anti-holomorphic involution (real structure) that keeps $D$ invariant. We show that the root stack $\mathfrak{X}_D$ naturally admits a real structure compatible with $X$. We also establish an equivalence of categories between the category of real logarithmic connections on this root stack and the category of real parabolic connections on $X$.

Real Structures on Root Stacks and Parabolic Connections

TL;DR

This work develops a real-structure framework for root stacks and parabolic connections. It proves that a smooth complex variety X with a real structure Οƒ and a real sncd D induces a compatible real structure on the r-th root stack 𝔛 of (π’ͺ_X(D), s_D), and establishes an equivalence between real parabolic vector bundles on (X, D) and real vector bundles on (𝔛, Οƒ_𝔛), as well as between real logarithmic connections on (𝔛, 𝔇) and real parabolic connections on (X, D) with weights in ; holomorphic real connections on 𝔛 correspond to real strongly parabolic connections on (X, D). The results unify real-structure considerations with Biswas-Borne correspondences and extend the parabolic-connection paradigm to real algebraic geometry, including a quotient-stack/Galois-cover perspective via Kawamata-type covers. Altogether, the paper provides a robust stack-theoretic bridge between real parabolic data and root-stack connections, enhancing tools for real nonabelian Hodge-type correspondences in higher dimensions.

Abstract

Let be a reduced effective strict normal crossing divisor on a smooth complex variety , and let be an associated root stack over . Suppose that admits an anti-holomorphic involution (real structure) that keeps invariant. We show that the root stack naturally admits a real structure compatible with . We also establish an equivalence of categories between the category of real logarithmic connections on this root stack and the category of real parabolic connections on .
Paper Structure (15 sections, 18 theorems, 101 equations)

This paper contains 15 sections, 18 theorems, 101 equations.

Table of Contents

  1. Introduction
  2. Real Structures
  3. Real structures on $\mathcal{O}_X$-modules
  4. Real Structures on Root Stacks
  5. Root stack
  6. Real structure on the tautological line bundle on root stack
  7. Biswas-Borne Correspondence
  8. Real structures on parabolic vector bundles
  9. Correspondence with real structure
  10. Real Connections On Root Stacks
  11. Connections on root stacks
  12. Real logarithmic connection and real parabolic connection
  13. Root stacks and ramified Galois covers
  14. Real structure on quotient stack
  15. Finite Galois coverings

Key Result

Theorem 1.0.1

There is an equivalence between the category of real parabolic vector bundles on $(X,\sigma)$ having weights in $\frac{1}{r}\mathbb{Z}$ and the category of real vector bundles on $(\mathfrak{X},\sigma_{\mathfrak{X}})$.

Theorems & Definitions (46)

  • Theorem 1.0.1: Theorem \ref{['thm:real-avatar-of-Biswas-Borne-correspondence']}
  • Theorem 1.0.2: Theorem \ref{['thm:real-log-parbolic-connection-correspondence']}
  • Definition 2.0.1
  • Definition 2.1.1
  • Remark 2.1.2
  • Definition 2.1.3
  • Proposition 3.1.1
  • proof
  • Remark 3.1.4
  • Proposition 3.1.5
  • ...and 36 more