Let $X$ be a smooth quasi-projective variety of dimension $n$ over the complex numbers (let us assume it is the complement of a normal crossings divisor in a smooth projective variety).

Then how does the mixed Hodge structure on cohomology with compact support relate to that of the usual cohomology.

By Poincare duality $H^k_c(X) \cong H^{2n-k}(X)$. But is this compatible with he mixed Hodge structures.

In my toy example $X$ is affine and $H^k(X)$ has pure Hodge structure of weight $2k$, can I figure out the weight of the Hodge structure on $H^k_c(X)$ from this?