Let $F$ be a smooth complete intersection of $r$ hypersurfaces of degree $d_{1},\dots,d_{r}$ in $\mathbb{P}^{n+r}$ over an algebraic closed field. A classical result of A. Roitman says that the group of zero-cycles of degree zero of $F$ is trivial: $$\text{CH}_{0}^{\text{deg 0}}(F)=0$$ if $\sum d_{i}\leq n+r$.

Now, for an equidimensional quasi projective scheme $X$, Bloch defines Higher Chow groups, see here for a quick introduction, $\text{CH}^{q}(X,m)$. Let $q=\dim F+n$ and consider $\text{CH}^{q}(F,n)$.

I have the following (very vague) questions:

• What should be the right analogue of the result given by Roitman in this context?
• Are there known results in this direction?

For example, is $\ker(\text{CH}^{q}(F,n)\to \text{CH}^{n}(k,n))$ finitely generated? Note that if $n=0$ and $\sum d_{i}\leq n+r$, we obtain the above result.

Thank you.

Let $F$ be a smooth complete intersection of $r$ hypersurfaces of degree $d_{1},\dots,d_{r}$ in $\mathbb{P}^{n+r}$ over an algebraic closed field. A classical result of A. Roitman says that the group of zero-cycles of degree zero of $F$ is trivial: $$\text{CH}_{0}^{\text{deg 0}}(F)=0$$ if $\sum d_{i}\leq n+r$.

Now, for an equidimensional quasi projective scheme $X$, Bloch defines Higher Chow groups, see here for a quick introduction, $\text{CH}^{q}(X,m)$. Let $q=\dim F+n$ and consider $\text{CH}^{q}(F,n)$.

I have the following (very vague) questions:

• What should be the right analogue of the result given by Roitman in this context?

• Are there known results in this direction?

  For example, is $\ker(\text{CH}^{q}(F,n)\to \text{CH}^{n}(k,n))$ finitely generated? Note that if $n=0$ and $\sum d_{i}\leq n+r$, we obtain the above result.

Thank you.