Sur l'injectivit\'e de l'application cycle de Jannsen

For specific classes of smooth, projective varieties $X$ over a field $k$, we compare two cycle maps on the torsion subgroup $CH^2(X)_{\text{tors} }$ of the second Chow group. The first one goes back to work of S. Bloch (1981), the second one is Jannsen's cycle map into continuous $\ell$-adic cohomology, whose injectivity properties have attracted attention in two recent papers. On the one hand, the comparison gives sufficient hypotheses to guarantee injectivity of Jannsen's cycle map sending $CH^2(X)$ to $H^4_{cont}(X, Z_{\ell}(2))$ on $\ell$-primary torsion. On the other hand, using counterexamples to injectivity of the first map due to Sansuc and the first author (1983), we give examples of smooth, projective, geometrically rational surfaces over a rational function field in one variable over a totally imaginary number field for which Jannsen's map for $\ell=2$ is not injective on $2$-torsion. This answers questions raised in a recent paper.