Abstract:
Let $X$ be the product of a smooth
projective curve $C$ and a smooth projective surface $S$ over a
field $K$. Assume the Chow group of zero-cycles on $S$ is just $Z$
over any algebraically closed field extension of $F$ (example :
Enriques surface). For $K$ the complex field, one may give
counterexamples to the integral Hodge conjecture for 1-cycles
(Benoist-Ottem) on $X$ and this may be understood from the point of
view of unramified cohomology. For $K$ a finite field, in joint work
with Federico Scavia (UBC, Vancouver) we give a simple condition on
$C$ and $S$ which ensures that the integral Tate conjecture holds
for 1-cycles on $X$. An equivalent formulation is a vanishing result
for unramified cohomology of degree 3.
