Genus of smooth varieties with small Chow group

Question

Let $X$ be a smooth projective variety over $\mathbb C$ with $d = \dim X \geq 1$. Let $CH(X)$ denotes the total Chow group of (cycles modulo rational equivalences of) $X$ and $CH(X)_{\mathbb Q} = CH(X)\otimes_{\mathbb Z} \mathbb Q$. My question is

Suppose $CH(X)_{\mathbb Q}$ is finite-dimensional as a $\mathbb Q$-vector space. When can we conclude that $H^d(X, \mathcal O_X) =0$?

I suspect that the answer might be always yes: in dimension one, such varieties have to be rational. In dimension two, I think it follows from a paper of Mumford. Also, it is true for flag manifolds.

More vaguely, I am also quite interested in:

Suppose $CH(X)_{\mathbb Q}$ is finite-dimensional as a $\mathbb Q$-vector space. What can we say about the geometry of $X$?

Any partial answers would be appreciated. Thanks!

Answer 1

As you suspect, the answer to your first question is yes.

A. Roĭtman in "Rational equivalence of zero-dimensional cycles". Math. USSR-Sb. 18 (1974), 571--588, generalised Mumford's theorem to show that if $h^q(X, \mathcal{O}_X) > 0$ for any $q>0$ then $CH_0(X)$ is infinite dimensional. If $q > 1$ then there is no surjection from the points of a variety over $\mathbb{C}$ onto $CH_0(X)$ so it is infinite dimensional in a very strong sense.

In general, it is conjectured -- generalizing Bloch's conjecture for surfaces -- that $CH(X)$ is finite dimensional iff $h^p(X, \Omega^q) = 0$ for $p \neq q$. In this case one expects that the cycle class map $CH(X) \to H(X)$ is an isomorphism, where $H(X)$ denotes the total singular cohomology with $\mathbb{Q}$-coefficients.

The difficult direction is proving finite dimensionality -- this is not even known for surfaces -- but I think the other direction is also not known in full generality. However, if the cycle class map is injective then Jannsen has proved the above vanishing of the Hodge numbers (in Motivic sheaves and filtrations on Chow groups. Motives (Seattle, WA, 1991), 245–302, Proc. Sympos. Pure Math., 55, Part 1, Amer. Math. Soc., Providence, RI, 1994.)

Answer 2

For a smooth proper complex variety $X$, the condition that $CH^*(X)_{\mathbb Q}$ is a finite dimensional vector space is equivalent to the Chow motive of $X$ being a direct sum of Tate motives. In particular, the singular cohomology of $X$ (with $\mathbb Q$ coefficients) coincides with $CH^*(X)_{\mathbb Q}$ (and thus the Hodge decomposition is rather trivial). This can be proved using the technique of decomposition of the diagonal.

A version of this argument (for an arbitrary Chow motive) appears in an article of Kimura:

MR2562457 (2010j:14014), Surjectivity of the cycle map for Chow motives. in ``Motives and algebraic cycles'', 157–165, Fields Inst. Commun., 56, Amer. Math. Soc., Providence, RI, 2009.

Actually, one can even make do without the ``supported correspondences'', by reasoning as in Kapil Paranjape's article

MR1283872 (95g:14008), Cohomological and cycle-theoretic connectivity. Ann. of Math. (2) 139 (1994), no. 3, 641–660.

But they do give a clean way to write the proof.

There is a follow-up by Charles Vial (again, there is no real need to use ``birational motives'' in the proofs, for this result):

MR2738925 (2012c:14010), Pure motives with representable Chow groups, C. R. Math. Acad. Sci. Paris 348 (2010), no. 21-22, 1191–1195.