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.
