Let $X$ be a smooth compact orientable manifold (or variety) and let $j: U \subset X$ be the complement to a union $\bigcup_{i \in I} X_i$ of smooth compact orientable manifolds. Suppose that $A$ is a sheaf on $X$ - say, the locally constant sheaf $Z$ (although the question below can be asked for coherent sheaves too). Let $Z_U$ be $j_!j^* Z$, i.e. the locally constant sheaf on $U$ extended by zero to $X$. The usual set theoretic inclusion-exclusion formula leads to a long exact sequence of sheaves $0 \to Z_U \to Z \to \oplus Z_{X_i} \to \oplus Z_{X_i \cap X_j} \to \ldots$ where for a closed subset $f: W \subset X$ one sets $Z_W = f_* f^* Z$.

This leads to a spectral sequence with first page given by cohomology of finite intersections $X_{i_1} \cap \ldots \cap X_{i_s}$, and the differential is induced by the combinatorial inclusion-exclusion formula.

Are there any examples when the differential of $E_2$ is non zero and known explicitly (which means that we also know the E_2 terms)? Maybe something in terms of excess intersection bundles for intersections of $X_i$, some Gysin maps, etc/?