Meromorphic Hodge moduli spaces for reductive groups in arbitrary characteristic
Andres Fernandez Herrero, Siqing Zhang
TL;DR
The paper develops a comprehensive moduli theory for meromorphic $t$-connections on $G$-bundles over curves in arbitrary characteristic, introducing the meromorphic Hodge moduli stack $ ext{Hodge}^D_G$ and its semistable locus with an adequate moduli space $M_{ ext{Hod},G,d}^D$ quasi-projective over $\A^1_S$. It proves a theta-stratification framework for semistability, establishes semistable reduction, and shows smoothness of the semistable locus when pole divisors have nonempty fibers. In positive characteristic, it constructs the Hodge-Hitchin morphism and proves its properness under height/characteristic bounds, leveraging a Steinberg-type embedding and Chow-theoretic arguments. Collectively, these results extend Hitchin-type moduli to reductive groups in arbitrary characteristic and lay groundwork for ramified non-abelian Hodge theory in prime characteristic. The work provides a robust foundation for future cohomological and geometric Langlands developments in characteristic $p$ for general reductive groups.
Abstract
Fix a smooth projective family of curves $C \to S$ and a split reductive group scheme $G$ over a Noetherian base scheme $S$. For any (possibly nonreduced) fixed relative Cartier divisor $D$, we provide a treatment of the moduli of $G$-bundles on the fibers of $C$ equipped with $t$-connections with pole orders bounded by $D$. Under mild assumptions on the characteristics of all the residue fields of $S$, we construct a Hodge moduli space $M_{Hod, G} \to \mathbb{A}^1_S$ for the semistable locus, construct a Harder-Narasimhan stratification, and thus obtain a semistable reduction theorem. If all the fibers of the divisor of poles $D$ are nonempty, then we show that the stack of semistable objects is smooth over $\mathbb{A}^1_{S}$. We also define a Hodge-Hitchin morphism in positive characteristic and prove that it is proper.