# Poincare duality for mixed motives

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, $$H^n_{c}(U_{\overline{k}},\mathbb{Q}_{\ell})^{\vee} \simeq H^{2d-n}(U_{\overline{k}},\mathbb{Q}_\ell)(d),~d=\text{dim}\,X$$ 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 $$h^n_c(U) \simeq h^{2d-n}(U)(d)$$ Does anyone know any references?

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