Modul: MAT770 Oberseminar: Algebraische Geometrie

## Higher enhancements of Nori motives and realisation functors

Talk by Dr. Swann Tubach

Speaker invited by: Prof. Dr. Joseph Ayoub

**Date:** 31.10.23 **Time:** 13.15 - 14.45 **Room:** Y27H25

Let k be a field of characteristic zero and X a quasi-projective k-variety. The category of perverse Nori motives over X is an abelian category modelled on perverse sheaves but instead of having coefficients in \Q-vector spaces, they have stalks in the Tannakian category of motives constructed by Nori. They were constructed by Ivorra and S. Morel. Their work, together with the work of Terenzi provides the 6 operations for the derived category of perverse Nori motives. By adapting an argument due to Nori in the setting of constructible sheaves on the complex points of X, we show that the derived category of perverse Nori motives is the derived category of its constructible heart. This enables us to see each of the 6 operations as a right derived functor, which have natural higher categorical lifts. Thanks to the work of Drew, Gallauer and Robalo, the existence of those lifts gives us a comparison functor from the category of Voevodsky motives over X, compatible with the operations. Our arguments also work for mixed Hodge modules, providing a Hodge realisation of étale Voevodsky motives. If times permits, we will explain how to use higher categorical tools to expand perverse Nori motives, together with the 6 operations, from quasi-projective varieties to all finite type k-schemes, and even to all qcqs schemes of characteristic zero (for those, we do not extend all the operations).