5
$\begingroup$

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?

Improve this question
$\endgroup$
1
  • 2
    $\begingroup$ For the reasons explained in Dan's answer, $H^k_c(X)\cong H^{2n-k}(X)^*\otimes \mathbb{Q}(-n)$ as MHS. In general, the weights are compatible with what happens in etale cohomology (I'm not sure if that helps you). $\endgroup$ Commented Mar 21, 2014 at 15:43

1 Answer 1

Reset to default
6
$\begingroup$

${}$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 $2n-2k$.

Improve this answer
$\endgroup$

You must log in to answer this question.

Start asking to get answers

Find the answer to your question by asking.

Ask question

Explore related questions

See similar questions with these tags.