Untwisting the twisted character variety via a stacky curve

Abstract

Given a smooth projective curve $X$ over $\mathbb{C}$, we show that the twisted character variety of $X$ is equivalent to a component of the character variety of a stacky curve $\mathcal{X}$, constructed by adding an orbifold point to $X$. This allows to interpret the non abelian Hodge correspondence in degree $d$ over $X$ as a statement about the geometry of the Hodge moduli space of $\lambda$-connections on $\mathcal{X}$. Following Shende's approach (see arXiv:1411.4975), we exploit this interpretation to compute the weights of the tautological classes on the cohomology of the twisted character variety, by replacing $\mathcal{X}$ with a simplicial scheme $\Delta_{\mathcal{X}}$ obtained from a triangulation.