Speaker: Anubhav Nanavaty
Affiliation: Cornell University
Title of Talk: Weight Filtrations and the $K$-Theory of Varieties
Date: December 30, 2025
Time: 16:00:00 Hours
Venue: A-369

Abstract: Weight Filtrations are mysterious: they record some shadow of how a variety might be recovered from smooth and projective ones. Some of the information recorded by weight filtrations can be understood via the motivic measures they define, i.e. group homomorphisms from the Grothendieck ring of varieties. With Zakharevich’s discovery of the higher $K$ groups of the category of varieties, there is an ongoing project to understand these groups by lifting motivic measures (on the level of $K_0$) to so-called ``derived" ones, i.e. on the level of $K_i$ for all $i$. I will describe some of this work, which shows that if one closely studies how the Gillet-Soul\'e weight complex is constructed, then one can also obtain derived motivic measures to non-additive categories as well, such as the compact objects in the category of motivic spaces, along with that of compact objects in the motivic stable homotopy category. These new derived motivic measures allow us to answer questions in the literature, providing new ways to understand the higher $K$ groups of varieties. Further, they provide new invariants associated to certain classes of birational automorphisms of any algebraic variety over a characteristic $0$ field.