I think that the statement below about relative de Rham complex is true (am I wrong?) If it is the case, a reference would be very helpful. (I admit that the statement sounds somewhat technical and specialized, but I still hope that it might be standard and known to experts).
Let $X_1,X_2$ be a smooth complex analytic manifold. Let $F\subset X_1\times X_2$ be a closed complex analytic submanifold such that the projections $p_{1,2}\colon F\to X_{1,2}$ are submersions. For a point $a\in X_2$, let $C:=p_2^{-1}(a)\cap F$. For any $n\in \mathbb{Z}_{\geq 0}$ let us denote by $C^{(n)}$ the $n$th infinitesimal neighborhood of $C$ inside $F$, and by $i_n\colon C^{(n)}\subset F$ the natural closed imbedding. Let $\tilde C:=p_1(C)\subset X_1$. Let $p_1^{(n)}\colon C^{(n)}\to \tilde C^{(n)}$ denote the restriction of $p_1$.
Is it true that for any $n\in \mathbb{Z}_{\geq 0}$ the complex of sheaves on $F$ \begin{eqnarray} (p_1^{(n)})^{-1}(\mathcal{O}_{\tilde C})\to i_{n*}i_n^*p_1^*(\mathcal{O}_{X_1})\to i_{n-1*}i_{n-1}^*(\Omega^1_{F/X_1}\otimes_{\mathcal{O}_F}p_1^*\mathcal{O}_{X_1}) \to \\ i_{n-2*}i_{n-2}^*(\Omega^2_{F/X_1}\otimes_{\mathcal{O}_F}p_1^*\mathcal{O}_{X_1})\to \dots \end{eqnarray} is exact?
Here the differentials are induced by the relative de Rham differentials. $f^*$ denotes the pull-back in the category of coherent sheaves, while $f^{-1}$ denotes the pull-back in the category of sheaves of vector spaces. Also, by convention, in the above complex the terms $i_{n-q*}i_{n-q}^*(\dots)$ vanish for $n-q<0$.
