Logarithmic geometry and Frobenius

TL;DR

This paper develops a new category of $$-adic sheaves with weight filtrations on logarithmic schemes over finite fields, linking logarithmic geometry with monodromy-weight conjecture insights.

Contribution

It introduces an abelian category of $$-adic sheaves with weight filtrations on logarithmic schemes, inspired by mixed Hodge structures and Deligne systems.

Findings

Analyzes asymptotic behaviors of higher direct images.

Establishes connections to the monodromy-weight conjecture.

Provides simple cases illustrating the theory.

Abstract

Based on the logarithmic algebraic geometry and the theory of Deligne systems, we define an abelian category of $\ell$-adic sheaves with weight filtrations on a logarithmic scheme over a finite field, which is similar to the category of variations of mixed Hodge structure. We consider asymptotic behaviors and simple cases of higher direct images of objects of this category. This category is closely related to the monodromy-weight conjecture.