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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?

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    $\begingroup$ If I'm not mistaken the motive with compact support is defined in Voevodsky-Suslin-Friedlander's book (Annals of Math Studies 143). $\endgroup$ – François Brunault Apr 16 '18 at 5:50
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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.

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