The compatibility of the Gysin sequence with mixed Hodge structures

Question

Let $X$ be a compact complex $n$-manifold and $D$ be a smooth comdimension $1$ submanifold. Also let $U:= X\setminus D$ and $j$ be the inclusion of $U$ in $X$. Then it is well known that the following (long exact) sequence respects mixed Hodge structures:

$$ \cdots\to H^{k-2}(D)(-1)\stackrel{\gamma_k}{\to}H^k(X)\stackrel{j^*}{\to}H^k(U)\stackrel{\text{Res}}{\to}H^{k-1}(D)(-1)\to\cdots.$$

Here $\gamma_k$ is the Gysin map obtained as the Poincare dual of $H^{2n-k}(X)\to H^{2n-k}(D)$. And $\text{Res}$ is the residue map that knocks off a factor of type $\frac{dz}{z}$ from a form.

My question: Where does one needs compactness of the manifold/ variety $X$? If one were to work with non-compact $X$, would the above sequence respect the mixed Hodge structure?

Answer 1

Yes, it's fine also if $X$ is not compact. More generally you have for any variety $X$ and subvariety $Z$ a long exact sequence $$ \ldots \to H^k_c(X \setminus Z) \to H^k_c(X) \to H^k_c(Z) \to H^{k+1}_c(X \setminus Z) \to \ldots $$ of mixed Hodge structures, which gives your long exact sequence by Poincaré duality when $X$ is smooth and $Z$ a smooth divisor.