On the image of the second l-adic Bloch map

TL;DR

This paper investigates the second l-adic Bloch map for smooth projective threefolds, establishing canonical isomorphisms between algebraic representatives and third l-adic cohomology groups, with implications for rationally connected varieties.

Contribution

It introduces a canonical abelian variety called the second algebraic representative and links it to the third l-adic cohomology for certain threefolds, extending previous rational coefficient results.

Findings

Existence of a canonical abelian variety modeling third l-adic cohomology.

Canonical identifications for rationally chain connected threefolds.

Integral correspondences induce isomorphisms in stably rational cases.

Abstract

For a smooth projective geometrically uniruled threefold defined over a perfect field we show that there exists a canonical abelian variety over the field, namely the second algebraic representative, whose rational Tate modules model canonically the third l-adic cohomology groups of the variety for all primes l. In addition, there exists a rational correspondence inducing these identifications. In the case of a geometrically rationally chain connected variety, one obtains canonical identifications between the integral Tate modules of the second algebraic representative and the third l-adic cohomology groups of the variety, and if the variety is a geometrically stably rational threefold, these identifications are induced by an integral correspondence. Our overall strategy consists in studying -- for arbitrary smooth projective varieties -- the image of the second ell-adic Bloch map restricted to the Tate module of algebraically trivial cycle classes in terms of the "correspondence (co)niveau filtration". This complements results with rational coefficients due to Suwa. In the appendix, we review the construction of the Bloch map and its basic properties.