Real Structures on Root Stacks and Parabolic Connections

TL;DR

This paper demonstrates that root stacks associated with complex varieties with divisors inherit real structures compatible with the base variety, and establishes an equivalence between real logarithmic and parabolic connections.

Contribution

It introduces a natural real structure on root stacks and proves an equivalence of categories between real logarithmic and parabolic connections.

Findings

Root stacks inherit real structures compatible with the base variety.

An equivalence of categories between real logarithmic and parabolic connections is established.

The results facilitate the study of real structures in complex algebraic geometry.

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 $X$.