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Let $X$ be an integral scheme of finite type over a field $k$ of dimension $d$ and $U$ an open dense smooth subscheme.

Let $K\in D_{c}^{b}(X,\bar{\mathbb{Q}}_{l})$ be such that $K_{U}=\bar{\mathbb{Q}}_{l}[d]$.

Moreover, we assume that $X=\bigcup\limits_{\alpha\in\Lambda} S_{\alpha}$ where $S_{\alpha}$ is a locally closed subscheme and $\Lambda$ is a set, not necessarily finite, such that $i_{\alpha}^{*}K=IC_{S_{\alpha}}$.

Do we have that $K=IC_{X}$?

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This should fail already in the simplest case when $X = \mathbb A^1$ and $U$ is the complement of a point. Namely, the trivial local system on $U$ and the point admits several different gluings to a perverse sheaf on $X$. The precise gluing data you need is explained in Beilinson's "Gluing perverse sheaves", see also http://arxiv.org/abs/1002.1686

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I do this all over $\mathbb{C}$. By [BBD] this should not be a problem.

Assume $X=\mathbb{C}P^1$, $U=\mathbb{C}$, $S_1= U$, $S_0=X-U$. Let further $j_i:S_i\hookrightarrow X$ be the inclusion maps. Then $K:=j_{1!} \underline{\mathbb{C}}_{S_1}[1]\oplus j_{0!} \underline{\mathbb{C}}_{S_0}$ satisfies your condition above and is obviously not isomorphic to $\underline{\mathbb{C}}_X=IC_X$.

Here $\underline{\mathbb{C}}_X$ is the constant sheaf on $X$.

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