# perverse sheaves – Nearby cycles without a function

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:

1. 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 $$i^*mathbb{Q}_X$$?
2. 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.
3. This appears rather powerful. Has this technique been applied usefully somewhere? In other words, where can I read more?!