Two coniveau filtrations

Abstract

A cohomology class of a smooth complex variety of dimension n has coniveau at least c if it vanishes in the complement of a closed subvariety of codimension at least c, and it has strong coniveau at least c if it comes by proper pushforward from the cohomology of a smooth variety of dimension at most $n-c$. We show that these two notions differ in general, both for integral classes on smooth projective varieties and for rational classes on smooth open varieties.