Invariant vector bundles and Hitchin systems
Zakaria Ouaras, Hacen Zelaci
TL;DR
The paper addresses the moduli of semistable Γ-invariant vector bundles on a Galois cover π: X → Y of curves and develops the Γ-invariant Hitchin system. It extends the involution case to arbitrary finite Γ by computing deformation dimensions via Lefschetz-type formulas, constructing a Γ-equivariant Hitchin base 𝓦^θ, and describing the Hitchin fibers in terms of Γ-invariant line bundles on spectral curves (BNR-type) for smooth types. A smooth-type criterion using Yang diagrams identifies when spectral curves are generically smooth and shows that, in those cases, fibers are open subsets of Γ-invariant Picard varieties with dominant pushforward maps, giving a completely integrable system and connected moduli. The appendix recalls Seshadri’s correspondence between Γ-invariant bundles and parabolic bundles on the quotient, strengthening links between invariant and parabolic geometries and situating the results within the broader parabolic-BNR framework.
Abstract
Let $X\rightarrow Y$ be a Galois cover with Galois group $Γ$, where $X$ and $Y$ are smooth complex projective curve of genus $\geqslant 2$. In this paper, we study the moduli spaces of semistable $Γ-$invariant vector bundles on $X$ and classify their connected components. We also study the Hitchin systems on these moduli spaces and determine their fibers in the smooth case.