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Let $X$ be a smooth affine variety of dimension $n>2$ over $\mathbb{C}$. From the examples I have seen (admittedly very little) it seems to me that these varieties don't have torsion classes a little bit after degree $\frac{n+1}{2}$, where grading is by codimension.

$\mathbf{Question}_i$:For $i>\frac{n+1}{2}$ is is true that $CH^{i}(X)_{torsion}=0$.

Note that the answer is affirmative if $i=n$ since by a theorem of Bloch (see "Torsion algebraic cycles and a theorem of Roitman" ) $CH^n(X)[l]$ is isomorphic to $H^{2n-1}_{et}(X,\mathbb{Q}_l/ \mathbb{Z}_l[n])$ and etale cohomology groups of affine scheme over $\mathbb{C}$ (with torsion coefficient) vanishes after degree $n$.

I don't know the answer even in the case $i=n-1$. If someone knows an example where this fails please do tell. If this is hard I would be grateful if one could point me to the right literature where these kind of questions are studied.

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    $\begingroup$ I believe that is contradicted by Koll\'ar's "Trento examples". For $n\geq 3$, for every integer $e>0$, and for $d$ sufficiently large and divisible, for a very general degree $d$ hypersurface $Y$ in $\mathbb{P}^n_k$, the image of the pushforward map on Chow groups, $$\text{CH}_1(Y)\to \text{CH}_1(\mathbb{P}^n_k),$$ is contained in $e\cdot \text{CH}_1(\mathbb{P}^n_k)$. Thus, by the exact sequence of Chow groups, the open complement $X$ of $Y$ in $\mathbb{P}^n_k$ has $\text{CH}_1(X)$ a nonzero torsion group annihilated by $e$. $\endgroup$
    – Jason Starr
    Commented Jun 21, 2019 at 12:35
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    $\begingroup$ I checked, and Koll\'ar's examples only hold if $n$ equals $4$. However, the result for every $n$ is conjectured (still open) by Griffiths and Harris. So Koll'ar's examples give a negative answer when $n$ equals $4$, and the Griffiths-Harris conjecture implies a negative answer for every $n>3$. $\endgroup$
    – Jason Starr
    Commented Jun 21, 2019 at 12:41
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    $\begingroup$ There are examples of smooth affine rational varieties $X$ of dimension $p+2$ for any prime $p$ (over, say complex numbers) such that $CH_1(X)$ has $p$-torsion. You can see my paper `Stably free modules' in the Am. J. Math. in the 80's. $\endgroup$
    – Mohan
    Commented Jun 21, 2019 at 12:49
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    $\begingroup$ I remember now that we spoke about this when I visited. $\endgroup$
    – Jason Starr
    Commented Jun 21, 2019 at 12:53

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