# The four operations on perverse motives

## TL;DR

This paper constructs a category of perverse mixed motives using étale motives, establishing the four Grothendieck operations and demonstrating their compatibility with classical geometric and Hodge-theoretic constructions.

## Contribution

It introduces a new abelian category of perverse mixed motives with four operations and functorialities, linking motivic and classical intersection cohomology theories.

## Key findings

- Equivalence to Nori motives over Spec(k)
- Motivic realization of intersection cohomology
- Compatibility of motivic and Hodge-theoretic constructions

## Abstract

Let $k$ be a field of characteristic zero with a fixed embedding $\sigma:k\hookrightarrow \mathbb{C}$ into the field of complex numbers. Given a $k$-variety $X$, we use the triangulated category of \'etale motives with rational coefficients on $X$ to construct an abelian category $\mathscr{M}(X)$ of perverse mixed motives. We show that over $\mathrm{Spec}(k)$ the category obtained is canonically equivalent to the usual category of Nori motives and that the derived categories $\mathrm{D}^{\mathrm{b}}(\mathscr{M}(X))$ are equipped with the four operations of Grothendieck (for morphisms of quasi-projective $k$-varieties) as well as nearby and vanishing cycles functors and a formalism of weights. In particular, as an application, we show that many classical constructions done with perverse sheaves, such as intersection cohomology groups or Leray spectral sequences, are motivic and therefore compatible with Hodge theory. This recovers and strengthens work by Zucker, Saito, Arapura and de Cataldo-Migliorini and provide an arithmetic proof of the pureness of intersection cohomology with coefficients in a geometric variation of Hodge structures.