Séminaire d'arithmétique à Lyon
# The six functor formalism for perverse Nori motives

→
Europe/Paris

Description

Let $k$ be a subfield of the complex numbers. After Deligne, the

singular cohomology groups with rational coefficients of

quasi-projective $k$-varieties are more than just $\Q$-vector spaces,

since they carry a canonical $\Q$-mixed Hodge structure; all linear maps

between cohomology groups "coming from geometry" are automatically

compatible with this additional structure.

The theory of Nori motives provides a $\Q$-linear abelian category of

coefficients which further refines the category of $\Q$-mixed Hodge

structures: in fact, it is designed to have the finest possible

structure that singular cohomology groups of $k$-varieties can possibly

carry. Conjecturally, Nori's category should be the correct abelian

category of mixed motives over $k$ envisioned by Grothendieck,

Beilinson, Deligne and others; however, proving this seems to be totally

out of reach at present.

As conjectured by Beilinson, the theory of Mixed Motives should admit an

enhancement to a theory of Mixed Motivic Sheaves enjoying a complete six

functor formalism. In the last decade there have been several attempts

at extending Nori's original theory to a theory of "Nori motivic

sheaves", the hardest problem being precisely the construction of the

six functors.

After reviewing Nori's original theory in some detail, I will present

the theory of perverse Nori motives recently introduced by Ivorra and

Morel. A complete six functor formalism is now available in this

setting, by work of Ivorra--Morel and of myself; the final goal of my

talk is to sketch the main ideas behind this construction.