On the Nori and Hodge realisations of Voevodsky motives

TL;DR

This paper establishes a deep connection between Voevodsky motives, Nori motives, and mixed Hodge modules by constructing functors and equivalences that preserve key operations, advancing the understanding of motivic categories.

Contribution

It constructs $ta$-categorical lifts of six operations, defines realization functors, and proves equivalences between categories of motives, enhancing the structural understanding of motivic theories.

Findings

Derived categories of perverse Nori motives and mixed Hodge modules are the derived categories of their constructible hearts.

Constructed $ta$-categorical lifts of the six operations for motives.

Proved equivalences between categories of Artin motives and conditions for motivic $t$-structures.

Abstract

We show that the derived category of perverse Nori motives and mixed Hodge modules are the derived categories of their constructible hearts. This enables us to construct $\infty$-categorical lifts of the six operations and therefore to obtain realisation functors from the category of Voevodsky \'etale motives to the derived categories of perverse Nori motives and mixed Hodge modules that commute with the operations. We give a proof that the realisation induces an equivalence of categories between Artin motives in the category of \'etale motives and Artin motives in the derived category of Nori motives. We also prove that if a motivic $t$-structure exists then Voevodsky \'etale motives and the derived category of perverse Nori motives are equivalent. Finally we give a presentation of the indization of the derived category of perverse Nori motives as a category of modules in Voevodsky \'etale motives that gives a continuity result for perverse Nori motives.