Around algebraic equivalence of cycles

Question

Let $k$ be a finitely generated field, $X$ a smooth projective $k$-variety, $\ell$ a prime number, $\ell\in k^{\times}$, $r\ge 0$ an integer.

The Tate conjecture asserts surjectivity of the cycle class map:

$$c^r_{\ell}(X): Z_r(X)\otimes_{\mathbf{Z}}\mathbf{Q}_{\ell}\to H^{2r}_{\rm et}(X_{k^{\rm sep}},\mathbf{Q}_{\ell}(r))^{\text{Gal}(k^{\rm sep}/k)}$$

$c^r_{\ell}(X)$ factors through the group of $r$-cycles modulo algebraic equivalence, $\text{NS}^r(X)\otimes_{\mathbf{Z}}\mathbf{Q}_{\ell}$. We denote by $\text{ns}_{\ell}^r(X)$ the resulting cycle map.

• It would seem the Tate conjecture would then follow from surjectivity of $\text{ns}^r_{\ell}(X)$. Am I missing something up there? Chow groups of cycles modulo rational equivalence are much larger than Néron Severi groups, so this is saying Tate cycles are expected to all be images of cycles modulo algebraic equivalence. Is this correct?

• It is known (Thm. of the Base) that $\text{NS}^1(X)$ is finitely generated. Is this known for arbitrary $r\ge 0$?

• Is anything known about $\ker(\text{CH}^r(X)\twoheadrightarrow\text{NS}^r(X))$?

Answer 1

A summary of some of these results is given in §19.3 of Fulton's Intersection theory [Ful98]. To address for example the questions you ask:

• If $A \stackrel f\twoheadrightarrow B \stackrel g \to C$ are maps, then the images of $g$ and $gf$ agree. It doesn't matter that $B$ is 'smaller'.
• It is not true that $B^2(X) = \operatorname{CH}^2(X)/\!\sim_{\text{rat}}$ is finitely generated; in fact even $B^2(X) \otimes \mathbb Q$ need not be finite dimensional [Ful98, 19.3.3]; this is due to Clemens [Cle83]. The example is three-dimensional. It was not known at the time of writing of [Ful98] whether the torsion in $B^r(X)$ is finite; I don't know if this is known yet.
• The kernel of $\operatorname{CH}^r(X) \to B^r(X)$ can sometimes be given the structure of an abelian variety, but sometimes it's bigger than that [Ful98, 19.3.4]. Even for $r = \dim(X)$ (i.e. $0$-cycles), this group can be very big [Ful98, 19.3.5].

The notation $B$ for what you call $\operatorname{NS}$ seems to be standard, but possibly not the only standard.

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References.

[Cle83] Clemens, Herbert, Homological equivalence, modulo algebraic equivalence, is not finitely generated. Publ. Math. Inst. Hautes Étud. Sci. 58, p. 231-250 (1983). Available online through Numdam. ZBL0529.14002.

[Ful98] Fulton, William, Intersection theory. Second edition. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) 2. Springer-Verlag, Berlin, 1998. ZBL0885.14002.