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

TL;DR

This paper compares two cycle maps on the torsion subgroup of the second Chow group for certain varieties, establishing conditions for injectivity of Jannsen's map and providing counterexamples where it fails.

Contribution

It offers new criteria for the injectivity of Jannsen's cycle map and constructs explicit counterexamples over specific fields.

Findings

Sufficient conditions for Jannsen's map to be injective on torsion.

Counterexamples showing non-injectivity of Jannsen's map for 2-torsion.

Answers to recent questions about cycle map injectivity.

Abstract

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.