Let $X$ be a smooth algebraic variety (over some field of char 0), $Z$ a smooth closed subvariety of codimension 1, $i : Z \hookrightarrow X$ the inclusion, and $j : U \hookrightarrow X$ the complementary open immersion.
I've read that for any regular holonomic D-module $M$ on $U$, there is a distinguished triangle in $D_h(D_X)$, $$ j_! M \to j_* M \to i_* i^* j_* M \to [+1], $$ where $j_!$, $j_*$ are the direct images with and without compact supports.
If $M$ is given by a vector bundle with connection on $U$, and we can find an extension $\bar{M}$ of $M$ to a vector bundle on $X$ with a logarithmic connection along $Z$, can we use this to give a concrete description of the $D$-module $i^* j_* M$ on $Z$?
[Edit: I probably want to suppose the monodromy is unipotent, and $\bar{M}$ is Deligne's canonical extension of $M$; or some other condition that will allow me to say that $\bar{M} \otimes_{\mathcal{O}_X} \Omega^\bullet_X(\log Z)$ computes $R j_* (M \otimes_{\mathcal{O}_U} \Omega^\bullet_U)$, and similarly $\bar{M} \otimes_{\mathcal{O}_X} \Omega^\bullet_X(\log Z)(-Z)$ computes $Rj_! (M \otimes_{\mathcal{O}_U} \Omega^\bullet_U)$.]
I'd like it if one could identify the de Rham complex $DR^\bullet_Z(i^* j_* M)$ (in the bounded derived category of abelian sheaves on $Z$) with $i^* (\bar{M} \otimes_{\mathcal{O}_X} \Omega^\bullet_X(\log Z))$.
