Let $X$ be a smooth separated algebraic variety over $\mathbb{C}$ and $Z \subset X$ a subvariety of codimension $p$. There are no compactness assumptions. I am looking for an isomorphism of mixed hodge structures of the form
$(H^k(Z), W, F) \xrightarrow{\sim} (H^{k+2p}_Z(X), W, F) (p)$
I'm only aware of this statement in case $X$ is compact, as given in Fujiki's "Duality of Mixed Hodge structures", 1980.
More Info: I only need the case $k=0$ i.e. want that $H^{2p}_Z(X)\cong H^0(Z)$ has pure hodge structure of type $(p,p)$.
A Naive approach to check if the isomorphisms $ H^k(Z) \overset{Thom}\longrightarrow H^{k+2p}(N_{X/Z}, {N_{X/Z}-Z}) \cong H^{k+2p}_Z(X)$ give the iso of hodge structures fails at first glance: Peters-Steenbrink require either $X$ or $Z$ to be compact (Theorem 6.27 and following discussion) for the cup with the thom-class to give the desired morphism of mixed hodge structures.
