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?



mixed Hodge structure on Cohomology with compact support
${}$Hi Chitro. The Poincaré duality pairing $H^k(X) \otimes H^{2n-k}\_c(X) \to H^{2n}\_c(X)$ is a morphism of mixed Hodge structures. So in your example, for instance, $H^{2n-k}\_c(X)$ is pure of weight $2n-2k$. 


