On the invariant cycle theorem for families of Nori motives

TL;DR

This paper extends the invariant cycle theorem to mixed Nori motives, providing a motivic enhancement of Deligne's fixed part theorem, with simplified proofs and broader applicability in motivic Hodge theory.

Contribution

It introduces a motivic version of the invariant cycle theorem for mixed Nori motives, generalizing previous results and simplifying the proof framework.

Findings

Extended the invariant cycle theorem to mixed Nori motives.

Provided a simpler proof within the Nori motives framework.

Strengthened the connection between motivic and Hodge-theoretic fixed parts.

Abstract

In this paper, we prove a motivic enhancement of the theorem of the fixed part in Hodge theory due to Deligne. In the pure motivic case, this was done for the first time by Andr\'e in [And96]. Our main result is an extension to the mixed case, which strengthens a result by Arapura [Ara13] and also provides an alternative and simpler proof, in the framework of Nori motives, of a result by Ayoub [Ayo14].