# Special cases of the Kimura-O’Sullivan conjecture, i.e., examples of finite dimensional motives?

In this 2005 paper, Kimura introduces a notion of finite dimensionality for Chow motives, defined in terms of vanishing of high symmetric and wedge powers. Toward the end of his paper, he conjectures that the Chow motive $h(X)$ of any smooth variety $X$ is finite dimensional [Conjecture 7.1]. The (more general?) Kimura-O’Sullivan conjecture states that every Chow motive is finite dimensional.

In his paper, Kimura proves that Chow motives of smooth projective curves are finite dimensional [Corollary 4.4] and that tensor products of finite dimensional motives are finite dimensional [Corollary 5.11]. What is the current state of knowledge about Chow motives satisfying the Kimura-O’Sullivan conjecture? More specifically:

Q1. What other specific Chow motives are known to be finite dimensional?

Q2. What other constructions from finite dimensional Chow motives are known to return finite dimensional Chow motives?

Q3. I would expect that for any finite surjective map $f:X\longrightarrow Y$ of smooth proper $k$-schemes over a fixed field $k$, if the Chow motive $h(Y)$ is finite dimensional, then the Chow motive $h(X)$ is also finite dimensional. Is there an obvious way to see this? Is this known? (If I understand correctly, the dual statement minus the finiteness assumption — the statement that if $h(X)$ is finite dimensional and $f$ is surjective then $h(Y)$ is finite dimensional — is Proposition 6.9 in Kimura's paper.)

Q3. This is definitely not known at all! Any projective variety is a finite cover of a projective space, and $h(\mathbb P^n)$ is of course finite dimensional.
• @DanPetersen, This probably just betrays my lack of understanding of the category of Chow motives, but regarding Q3: are there special cases of surjective finite morphisms where we can say more? For instance, can we say more if $f:X\to Y$ is finite étale, or if $f$ is a $G$-torsor with $G$ a finite group? – Tyler Foster Apr 18 '17 at 7:09