Meromorphic Hodge moduli spaces for reductive groups in arbitrary characteristic

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 $M_{Hod,G}\to {\mathbb{A}}_S^1$ 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 ${\mathbb{A}}_S^1$. We also define a Hodge-Hitchin morphism in positive characteristic and prove that it is proper.

Accepted version. To appear in Mich. Math. J