# Regular logarithmic connections

## TL;DR

This paper introduces a new concept in algebraic geometry called regular integrable connections and relates them to cohomology theories.

## Contribution

The paper introduces regular integrable connections on log schemes and establishes a new equivalence with integrable connections on their analytifications.

## Key findings

- An equivalence is constructed between regular integrable connections and integrable connections on the analytification.
- The work extends Deligne's results and provides a topological description using constructible sheaves on the Kato–Nakayama space.
- The use of canonical extensions and good compactifications is essential for the results.

## Abstract

We introduce the notion of a regular integrable connection on a smooth log scheme over \documentclass[12pt]{minimal}
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\begin{document}$$\textbf{C}$$\end{document}C and construct an equivalence between the category of such connections and the category of integrable connections on its analytification, compatible with de Rham cohomology. This extends the work of Deligne (when the log structure is trivial), and combined with the work of Ogus yields a topological description of the category of regular connections in terms of certain constructible sheaves on the Kato–Nakayama space. The key ingredients are the notion of a canonical extension in this context and the existence of good compactifications of log schemes obtained recently by Włodarczyk.