Meromorphic Hodge moduli spaces for reductive groups in arbitrary characteristic

TL;DR

This paper constructs and studies a moduli space of G-bundles with t-connections on families of curves, establishing properties like smoothness, stratification, and properness of associated morphisms in arbitrary characteristic.

Contribution

It introduces a Hodge moduli space for G-bundles with t-connections in arbitrary characteristic, including stratification and properness results.

Findings

Constructed a Hodge moduli space for G-bundles with t-connections.

Proved the smoothness of the stack of semistable objects under certain conditions.

Established the properness of the Hodge-Hitchin morphism in positive 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 $M_{Hod, G} \to \mathbb{A}^1_S$ 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.