A cohomology class of a smooth complex variety of dimension $n$ has coniveau $geq c$ if it vanishes in the complement of a closed subvariety of codimension $geq c$, and has strong coniveau $geq c$ if it comes by proper pushforward from the cohomology of a smooth variety of dimension $leq n-c$. 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.
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