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Let $D$ be a divisor in a (compact) Kahler manifold $X$. Let $N$ be the total space of the normal bundle of $D$ in $X$ and identify $D$ with the zero section of $N$. Let $m\subset O_{X}$ and $p\subset O_{N}$ denote the ideal sheaves of $D$ in $X$ and $N$ respectively. Here $O_{X}$ and $O_{N}$ are the structure sheaves of $X$ and $N$ respectively. Then, by definition, $(D, O_{X}/m^{k})$ and $(D, O_{N}/p^{k})$ are the $k$-th infinitesimal neighbourhoods of $D \subset X$ and $D\subset N$ respectively.

My question is as follows. Suppose I have an isomorphism $\psi_{l}:(D, O_{X}/m^{l})\rightarrow (D, O_{N}/p^{l})$ between the $l$-th infinitesimal neighbourhoods, for some $l\geq 1$. Consider the $l$-jets along $D$ of diffeomorphisms fixing $D$ between some small tubular neighbourhood of $D$ in $X$ and $D$ in $N$. Does the isomorphism $\psi_{l}$ between the $l$-th infinitesimal neighbourhoods induce in some way a natural element of this space of $l$-jets?

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