Let $Z\subset U$ be a Kähler tubular neighborhood of a compact manifold $Z$ of codimension $r$. Consider de Rham complexes of smooth differential forms $\Lambda^{*,*}(Z),\Lambda^{*,*}(U)$, let $\Lambda_c^{*,*}(U)$ denote de Rham complex of compactly supported forms.
The Gysin isomorphism $g$:$$H^*(\Lambda^*(Z))\simeq H^*(\Lambda^*(U))\overset{g}{\longrightarrow} H^{*+2r}(\Lambda^*(U))\simeq H^{*+2r}(U,U-Z)$$ is defined over $\mathbb{Z}$.
Say that Gysin homomorphism is realizible if there is $\eta\in \Lambda^{r,r}(U)$ such that $\Lambda^{*,*}(U)\overset{\eta\wedge-}{\longrightarrow} \Lambda^{*+r,*+r}_c(U)$, given by multiplication by $\eta$, induces $g$ on cohomology.
$\textbf{Q1}$ Is it true that the the Gysin homomorphism is always realizible?
If $Z$ happens to be the zero locus of a regular section $s\in \Gamma(U,E)$ of a hermitian holomorphic vector bundle $E$ over $U$, which is trivial over $U-Z$, then I guess that the Gysin homorphism is realizible: one can represent class $c_r(E)$ through hermitian cuvature of type $(1,1)$ by a well-known formula, which will produce a representative of $c_r(E)$ by $(r,r)$-form with compact support. The particular case of such construction is the Poincare-Lelong formula for line bunles.
$\textbf{Q2}$ Is it true that $U$ admits such vector bundle $E$ in case $r>1$?
The cohomology group $H^*(U)=H^*(Z)$ naturally admits a pure Hodge structure. One can consider the Gysin isomorphism $g:H^*(Z)\to H^{*+2r}(U,U-Z)$ as a morphism of pure Hodge structures of degree $r$.
$\textbf{Q3}$ Is it possible to construct a natural filtration $F$ on $\Lambda_c^{*,*}(U)$ such that the induced filtration on cohomology $H^*(U,U-Z)\simeq H^{*-2r}(Z)(-r)$ gives the Hodge filtration?
Note that $\partial\bar{\partial}$-lemma does not hold for the de Rham complexes above and I do not require that the associated spectral sequence degenerate at ${}_{F}E_1$. For an illustration one can consider the same question for $\Lambda^{*,*}(U)$. In this case naive filtration by number of $dz$'s $\textit{does not}$ in general induce the Hodge filtration on $H^*(U)=H^*(Z)$. Indeed, this is equivalent to a question when a closed $(p,q)$-form on $Z$ extends to a closed $(p,q)$-form on $U$. A counter-example is provided by smooth cubic $C$ in $\mathbb{P}^2$, where global $(1,0)$-closed form on $C$ does not extends on any tubular neighborhood. On the other hand one can always take preimage of the Hodge filtration on $\Lambda^{**}(Z)$ via restriction $\Lambda^{**}(U)\to \Lambda^{**}(Z)$.
