Suppose that:

- $X$ is a smooth complex algebraic variety,
- $f : X to D$ is a map to a small disc, smooth away from 0,
- $Z_epsilon = f^{-1}(epsilon)$, and $Z = Z_0$.

Then there is a procedure (“nearby cycles”) which produces a complex $psi_f(mathbb{Q}_X)$ on $Z$ whose cohomology agrees with that of $Z_epsilon$ for $epsilon ne 0$. (My notation is that $mathbb{Q}_X$ is always shifted so as to be perverse.)

This belongs to a powerful toolbox of techniques to relate the cohomology of general and special fibres, with important applications in algebraic geometry, number theory and representation theory.

**Question:** Suppose that I just give you $Z subset X$ but not $f$, can I “guess” $psi_f(mathbb{Q}_X)$?

Here is a rough proposal for how to do so, and I’m wondering if this is discussed somewhere in the literature. It seems a little like magic, and I might be making mistakes.

Firstly, $psi_f(mathbb{Q}_X)$ is a perverse sheaf, and it comes with a monodromy endomorphism $mu$. **I assume that $mu$ is unipotent**. Hence $N = 1-mu$ is a nilpotent endomorphism of the nearby cycles, and I have a short exact sequence of perverse sheaves:

$$ 0 to phi_f(mathbb{Q}_X) stackrel{N}{to} psi_f(mathbb{Q}_X) to i^*mathbb{Q}_X to 0$$

Thus, $i^*mathbb{Q}_X$ is the “coinvariants of the monodromy”.

Secondly, $psi_f(mathbb{Q}_X)$ carries a weight filtration $W$, and a deep theorem of Gabber states that the weight filtration agrees with the monodromy filtration. In particular:

$$N^i : gr_W^{-i}(psi_f(mathbb{Q}_X)) stackrel{sim}{to} gr_W^{i}(psi_f(mathbb{Q}_X))$$

is an isomorphism.

Now, assume that I know the successive subquotients of the weight filtration on $i^*mathbb{Q}_X$. Then, it seems that the above gives a very good picture of the associated graded of the weight filtration on $psi_f(mathbb{Q}_X)$. Namely, every $IC_lambda$ which occurs in $gr_W^{-i}(i^*mathbb{Q}_X$) contributes an $IC_lambda$ in weight filtration steps $-i, -i+2, dots, i-2, i$ to the weight filtration on $psi_f(mathbb{Q}_X)$.

An analogy: any finite-dimensional representation of $mathfrak{sl}_2(mathbb{C})$ is recoverable from its lowest weight vectors. Under this analogy, the lowest weight vectors are given by $i^*mathbb{Q}_X$.

Some precise questions:

- Under the above assumptions is it correct that I can recover the associated graded of the weight filtration on $psi_f(mathbb{Q}_X)$ from that of $i^*mathbb{Q}_X$?
- This appears to imply that $psi_f(mathbb{Q}_X)$ is automatically constructible for any stratification that makes $i^*mathbb{Q}_X$ constructible, which surprises me a little.
- This appears rather powerful. Has this technique been applied usefully somewhere? In other words, where can I read more?!