Mixed Hodge structures with modulus

TL;DR

This paper introduces a generalized notion of mixed Hodge structures with modulus, extending classical and level one Hodge structures, and connects these to Laumon 1-motives to generalize Albanese varieties with modulus.

Contribution

It defines mixed Hodge structures with modulus for arbitrary dimension and links them to Laumon 1-motives, broadening the scope of Hodge theory and Albanese varieties with modulus.

Findings

Established a new category of mixed Hodge structures with modulus.

Proved an equivalence between level one mixed Hodge structures with modulus and Laumon 1-motives.

Generalized Albanese varieties with modulus to include 1-motives.

Abstract

We define a notion of mixed Hodge structure with modulus that generalizes the classical notion of mixed Hodge structure introduced by Deligne and the level one Hodge structures with additive parts introduced by Kato and Russell in their description of Albanese varieties with modulus. With modulus triples of any dimension we attach mixed Hodge structures with modulus. We combine this construction with an equivalence between the category of level one mixed Hodge structures with modulus and the category of Laumon $1$-motives to generalize Kato-Russell's Albanese varieties with modulus to $1$-motives.