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Let $X$ be a $d$-dimensional connected smooth variety over $\overline{\mathbb{F}}_q$.

It is well known that if $X$ is isomorphic to an affine space, then all the $\ell$-adic compactly supported cohomology groups $H^i_c(X,\overline{\mathbb{Q}}_{\ell})$ vanish except $i=2d$. On the other hand, Borel proved the converse when $X$ is a homogeneous space of some connected affine algebraic group (see Theorem 1.4 here) in 1985.

I am wondering if it is known that the converse is true without the assumption on homogeneous, or is there any counterexample?

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    $\begingroup$ I think the counterexamples in the answer to mathoverflow.net/questions/7603/… should also be counterexamples for your question. (Leaving this as a comment and not an answer because I didn't think through the details of switching between $\mathbb{C}$ and $\overline{\mathbb{F}_p}$ for large $p$, but it should be doable.) $\endgroup$
    – David E Speyer
    Commented Feb 25, 2016 at 21:36
  • $\begingroup$ David, yes, I think it's doable also. After spreading the variety out over a finitely generated ring, you can use the comparison theorem + proper base change. $\endgroup$
    – Donu Arapura
    Commented Feb 25, 2016 at 22:59
  • $\begingroup$ The hard part is showing that the characteristic p fibers don't become isomorphic to affine space. I think that will require actually reading the references. $\endgroup$
    – David E Speyer
    Commented Feb 26, 2016 at 12:52
  • $\begingroup$ Thanks a lot. It seems in the counter-example there will be some problem on char=2 (as 2 appears in the denominator of the construction of the group action) ? $\endgroup$
    – user148212
    Commented Feb 26, 2016 at 13:04
  • $\begingroup$ @user148212 Yes, and even further I think people are suggesting that you may have to throw away a larger finite set of primes. $\endgroup$
    – Will Sawin
    Commented Feb 26, 2016 at 14:56

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