Real parabolic vector bundles over a real curve

Abstract

We define real parabolic structures on real vector bundles over a real curve. Let (X, σX) be a real curve, and let S ⊂ X be a non-empty finite subset of X such that σX(S) = S. Let N ≥ 2 be an integer. We construct an N-fold cyclic cover p : Y→ X in the category of real curves, ramified precisely over each point of S, and with the property that for any element g of the Galois group Γ, and any y ∈ Y, one has \(\sigma_Y(gy) = g^{-1}\sigma_Y(y)\). We established an equivalence between the category of real parabolic vector bundles on (X, σX) with real parabolic structure over S, all of whose weights are integral multiples of 1/N, and the category of real Γ-equivariant vector bundles on (Y, σY).