# Logarithmic ramifications of \'etale sheaves by restricting to curves

## TL;DR

This paper demonstrates that the Swan conductor of an étale sheaf on a smooth variety, as defined by logarithmic ramification theory, can be computed via classical Swan conductors on curves, extending previous results.

## Contribution

It extends the computation of Swan conductors from rank 1 sheaves to higher ranks using logarithmic ramification theory and applies this to generalize semi-continuity properties.

## Key findings

- Swan conductor can be computed via restriction to curves.
- Extension of results from rank 1 to higher rank sheaves.
- Logarithmic ramification theory aligns with classical Swan conductors on curves.

## Abstract

In this article, we prove that the Swan conductor of an \'etale sheaf on a smooth variety defined by Abbes and Saito's logarithmic ramification theory can be computed by its classical Swan conductors after restricting it to curves. It extends the same result for rank 1 sheaves due to Barrientos. As an application, we give a logarithmic ramification version of generalizations of Deligne and Laumon's lower semi-continuity property for Swan conductors of \'etale sheaves on relative curves to higher relative dimensions in a geometric situation.