TITLE: Genus of smooth varieties with small Chow group
QUESTION [11 upvotes]: 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!

REPLY [7 votes]: 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.