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



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> Suppose $CH(X)\_{\mathbb Q}$ is finite-dimensional as a $\mathbb Q$-vector space. When can we conclude that $H^d(X, \mathcal O\_X) =0$?
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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](https://projecteuclid.org/journals/kyoto-journal-of-mathematics/volume-9/issue-2/Rational-equivalence-of-0-cycles-on-surfaces/10.1215/kjm/1250523940.full). Also, it is true for flag manifolds.


More vaguely, I am also quite interested in:



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> Suppose $CH(X)\_{\mathbb Q}$ is finite-dimensional as a $\mathbb Q$-vector space. What can we say about the geometry of $X$?
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Any partial answers would be appreciated. Thanks!



Genus of smooth varieties with small Chow group
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.)


