John Ottem: Two coniveau filtrations

Abstract

A cohomology class of a smooth complex variety of dimension n is said to be of "coniveau" at least c if it vanishes on the complement of a closed subvariety of codimension at least c, and of "strong coniveau" at least c if it comes by proper pushforward from the cohomology of a smooth variety of dimension at most n–c. These notions give rise to two filtrations on the cohomology groups of a variety, which are known to coincide in many cases (for instance, they agree on the rational cohomology of any smooth projective variety). However, we show that they differ in general, both for integral classes on smooth projective varieties and for rational classes on smooth open varieties. The difference between the two filtrations also give rise to new birational invariants. In this talk, I will present some new examples where the two filtrations differ.