Projective Completion of Moduli of $t$-Connections on Curves in Positive and Mixed Characteristic

TL;DR

This paper extends Simpson's compactification method for moduli of t-connections on curves to positive and mixed characteristic settings, establishing projectivity results for these moduli spaces over various base rings.

Contribution

It generalizes Simpson's technique to a broader base ring context and demonstrates projectivity of moduli spaces in positive characteristic.

Findings

Established projectivity of Hodge, de Rham, and Dolbeault moduli spaces in positive characteristic.

Extended compactification techniques to universally Japanese base rings.

Provided new tools for studying moduli of t-connections in mixed characteristic.

Abstract

We generalize a compactification technique due to C. Simpson in the context of $\mathbb{G}_m$-actions over the ground field of complex numbers, to the case of a universally Japanese base ring. We complement this generalized compactification technique so that it can sometimes yield projectivity results for these compactifications. We apply these projectivity results to the Hodge, de Rham, and Dolbeault moduli spaces for curves, with special regards to ground fields of positive characteristic.