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This seems like a really basic question, but I somehow don't know and haven't been able to find the answer.

I suspect that (at least under suitable assumptions) there should be a relation between the following two constructions, but I'm looking for a precise theorem.

If I have a family $X$ over a disk, which I'm thinking of as a degeneration of smooth fibers $X_t$ to a singular fiber $X_0$, then there are two natural perverse sheaves I can cook up on $X_0$: one is the intersection cohomology of $X_0$, and the other is the nearby cycles of the intersection cohomology (which is constant) on $X-X_0$. What is the relation between these two?

To make things a little more precise: let $\eta \rightarrow S \leftarrow s$ be a henselian trait, $X \rightarrow S$ a map with $X_s$ and $X_{\eta}$ the special and generic fibers. I'm imagining that $X_{\eta} \rightarrow \eta$ is smooth, but perhaps this is unnecessary. How can I describe $IC(X_s, \mathbb{Q}_{\ell}))$? in terms of $\Psi(IC(X_{\eta}, \mathbb{Q}_{\ell}))$? (I expect an answer to involve the weight filtration and monodromy operator on nearby cycles, in addition to the "bare sheaf".)

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Let me give a quick and lazy answer. Throughout all statements are up to shift in the derived category.

Let us assume that $X$ and $f : X \to \eta$ is smooth. Then the nearby cycles sheaf $\Psi_f \mathbb{Q}_X$ is generically a constant sheaf (generically the "nearby cycles" are points, in other words generically the vanishing cycles sheaf is trivial). Now Gabber's theorem identifies the weight filtration on $\Psi_f \mathbb{Q}_X$ with the monodromy filtration. It follows that $\Psi_f \mathbb{Q}_X$ has a filtration such that one subquotient of this (i.e. $W_0/W_{-1}$) is (geometrically) semi-simple and contains $IC(X_0)$ as a summand. Thus, in a certain sense the intersection cohomology is "right in the middle" of the nearby cycles.

This has various concrete consequences. For example, one has a spectral sequence with $E_1$ page the successive subquotients of the weight filtration, which converges to the cohomology of the nearby cycles = cohomology of a smooth fibre. In general it is tricky to work out what the other subquotients will be (metaphorically this is akin to calculating which summands occur in the Decomposition Theorem). However in certain cases (e.g. semi-stable reduction) one can be precise. For more on these topics look at the Rapoport-Zink and Clemens-Schmid spectral sequences.

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