The four operations on perverse motives

Abstract

Let $k$ be a field of characteristic zero with a fixed embedding $\sigma :k\hookrightarrow \mathbf{C}$ into the field of complex numbers. Given a $k$-variety $X$, we use the triangulated category of étale motives with rational coefficients on $X$ to construct an abelian category $\mathcal{M}(X)$ 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 ${\mathrm{D}}^{\mathrm{b}}(\mathcal{M}(X))$ 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 provides an arithmetic proof of the pureness of intersection cohomology with coefficients in a geometric variation of Hodge structures.