A moduli space of character sheaves
Gabriel Ribeiro
TL;DR
The paper develops a de Rham-theoretic moduli theory for character sheaves on commutative connected groups, introducing the moduli space $G^{\flat}$ to parameterize multiplicative line bundles with integrable connection. It builds this theory through Cartier duality for commutative group stacks, Breen–Deligne resolutions, and the Laumon dual, relating $G^{\flat}$ to the de Rham space $G_{\mathrm{dR}}$ and to extension data $G^{\natural}=underline{Ext}^1(G_{\mathrm{dR}},\mathbb{G}_m)$; it provides exact sequences, dimension bounds, and functoriality results. The authors show that $G^{\flat}$ serves as a coarse moduli space for character sheaves, with explicit descriptions of its $k$-points and a precise decomposition in terms of the abelian, toric, and unipotent components, along with a full analysis of how these moduli compare to the Betti/analytic side, yielding isomorphisms in the semiabelian case via Riemann–Hilbert. The work connects generalized 1-motives, Laumon duality, and non-abelian Hodge-theoretic perspectives to provide a robust framework for understanding de Rham character sheaves and their moduli.
Abstract
We study de Rham character sheaves on a commutative connected algebraic group $G$, defined as multiplicative line bundles with integrable connection. We construct a group algebraic space $G^\flat$ representing their moduli problem on seminormal test schemes, and we investigate its functoriality and geometry. The main technical ingredient is a study of extension sheaves on the de Rham space $G_\text{dR}$. An appendix provides self-contained, elementary proofs of basic results on de Rham spaces that may be of independent interest.