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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.)

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Q1 + Q2: At present the Chow motives known to be finite dimensional are precisely those that are contained in the thick tensor subcategory generated by motives of abelian varieties. That is, the motives that can be obtained from motives of abelian varieties by tensorial operations, extensions, quotients and subobjects.

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.

That said, there is recent dramatic progress in this area. According to a preprint of Ayoub ("Topologie feuilletée et conservativité des réalisations classiques en caractéristique nulle", available on his webpage), the conservativity conjecture in characteristic zero may be within reach (but see the disclaimers in his preprint). This would in particular imply finite dimensionality for all Chow motives which have only even or odd cohomology.

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    $\begingroup$ Thank you for this interesting reference! Sorry, I have not read the text yet; still I would like to ask you whether are you sure that this result would imply that all Chow motives are Kimura-finite and not just Schur-finite? $\endgroup$ – Mikhail Bondarko Apr 17 '17 at 17:38
  • $\begingroup$ You're right, sorry - I was hasty. $\endgroup$ – Dan Petersen Apr 17 '17 at 19:48
  • $\begingroup$ @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? $\endgroup$ – Tyler Foster Apr 18 '17 at 7:09
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    $\begingroup$ No, I don't see why that should be the case. Look at a K3 which is an étale double cover of an Enriques surface - then we know finite dimensionality downstairs but not in general upstairs. $\endgroup$ – Dan Petersen Apr 18 '17 at 8:00

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