Remarques sur la cohomologie des groupes k\"ahl\'eriens nilpotents

TL;DR

This paper demonstrates that the cohomology groups of (virtually) nilpotent K"ahler groups possess a natural mixed Hodge structure, and explores how this structure influences known examples of such groups.

Contribution

It establishes that cohomology groups of nilpotent K"ahler groups have a natural mixed Hodge structure, extending understanding of their geometric and algebraic properties.

Findings

Cohomology groups of nilpotent K"ahler groups carry a mixed Hodge structure.

Hopf morphisms are compatible with mixed Hodge structures.

Illustration with known examples of non-abelian nilpotent K"ahler groups.

Abstract

In this note, we show that the cohomology groups of (virtually) nilpotent K\"ahler groups are naturally endowed with a mixed Hodge structure. These structures make the Hopf morphisms into mixed Hodge structures morphisms. We illustrate this fact with the study of known examples of non-abelian nilpotent K\"ahler groups.