  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.
