Suppose $k$ is a field of characteristic zero (and we assume it is a number field if necessary). If $U$ is a smooth quasi-projective variety over $k$, then there is Poincare duality,
\begin{equation}
H^n\_{c}(U\_{\overline{k}},\mathbb{Q}\_{\ell})^{\vee} \simeq H^{2d-n}(U\_{\overline{k}},\mathbb{Q}\_\ell)(d),~d=\text{dim}\,X
\end{equation}
where $H^n\_{c}(U\_{\overline{k}},\mathbb{Q}\_{\ell})$ is compactly supported etale cohomology group. 


**Question 1**: is it expected that there is a **mixed motive** $h^n\_c(U)$ whose $\ell$-adic realization is $H^n\_{c}(U\_{\overline{k}},\mathbb{Q}\_{\ell})$? I have not found references about "compactly supported mixed motives", could anyone give references on this?


**Question 2**: is it expected that there is a Poincare duality of the form 
\begin{equation}
h^n\_c(U) \simeq h^{2d-n}(U)(d)
\end{equation}
Does anyone know any references? 



Poincare duality for mixed motives
Yes, Voevodsky-Suslin-Friedlander's book is the first reference on the subject; you will find motives with compact support in section 4 of "Triangulated categories of motives over a field". However, this book says nothing about realizations. Thus you should consult some papers of Ayoub or <https://www.cambridge.org/core/journals/compositio-mathematica/article/etale-motives/7B4B764D3B84964B7E6D936A4F932977> for the theory of exceptional images of motives and their relation to realizations. Lastly, I don't expect that you will find much material on mixed motives since they are "too conjectural"; hence you would have to deduce the duality in the form you want from the conjectural properties of the motivic t-structure.


