On a motivic interpretation of primitive, variable and fixed cohomology

TL;DR

This paper explores the motivic interpretation of classical cohomological results like Lefschetz's primitive decomposition and the splitting of cohomology into fixed and variable parts for smooth projective varieties.

Contribution

It provides a motivic perspective on fundamental cohomological decompositions, linking classical results to motivic theory.

Findings

Motivic interpretation of primitive decomposition

Motivic splitting of cohomology into fixed and variable parts

Enhanced understanding of classical cohomological results

Abstract

This note addresses the motivic nature of some classical cohomological results due to Lefschetz, namely the primitive decomposition (for the cohomology of smooth projective varieties), and, secondly, the splitting of the cohomology of a complete intersection into the "fixed" and "variable part".