Logarithmic de Rham Stacks and Non-Abelian Hodge Theory

TL;DR

This paper introduces the logarithmic de Rham stack for pairs (X, D) in positive characteristic, establishing new versions of logarithmic Cartier descent and a logarithmic non-abelian Hodge theorem for curves, extending previous results.

Contribution

It develops the concept of the logarithmic de Rham stack and proves new logarithmic non-abelian Hodge theorems using a logarithmic Frobenius twist, advancing the theory in positive characteristic.

Findings

Proves a new logarithmic Cartier descent theorem.

Establishes a logarithmic non-abelian Hodge theorem for curves.

Extends previous logarithmic Hodge results of de Cataldo-Zhang.

Abstract

In this article, we introduce the logarithmic de Rham stack of a pair (X, D), for a smooth variety X over a field k of positive characteristic p, and D a strict normal crossings divisor on X. Using this stack, we prove a new version of logarithmic Cartier descent, and a new logarithmic non-abelian Hodge theorem for curves, both stated using a certain logarithmic Frobenius twist. Our logarithmic non-abelian Hodge theorem implies an earlier logarithmic non-abelian Hodge theorem of de Cataldo-Zhang.