Real structures on root stacks and parabolic connections

Let D be a reduced effective strict normal crossing divisor on a smooth complex variety X, and let \(\mathfrak {X}_D\) ${\mathfrak{X}}_D$ be the associated root stack over \(\mathbb C\) $\mathbb{C}$ . Suppose that X admits an anti-holomorphic involution (real structure) that keeps D invariant. We show that the root stack \(\mathfrak {X}_D\) ${\mathfrak{X}}_D$ 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.