# Nearby cycles without a function

Suppose that:

• $$X$$ is a smooth complex algebraic variety,
• $$f : X \to D$$ is a proper 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” $$\psi_f(\mathbb{Q}_X)$$? Put more simply, how much about the cohomology of a nearby smooth fibre $$Z_\epsilon$$ can I deduce from $$Z$$?

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:

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

Thus, $$i^*\mathbb{Q}_X$$ is the “invariants 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 highest weight vectors. Under this analogy, the highest weight vectors are given by (the associated graded for the weight filtration on) $$i^*\mathbb{Q}_X$$.

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?!

Another somewhat metaphorical way of phrasing this: suppose we have a semi-stable degeneration. Then the Rapoport-Zink spectral sequence gives us a recipe for computation which doesn't know about $$f$$. I am asking whether the terms of (analogue of the) the Rapoport-Zink spectral sequence can be read off from $$i^*\mathbb{Q}_X$$. (Here "analogue of the Rapoport-Zink spectral sequence" means the spectral sequence coming from the weight filtration on $$\psi_f(\mathbb{Q}_X)$$.)

• For 2 one has to be a bit careful: depending on how you define stratification, a perverse sheaf can be constructible for a stratification without its associated graded being constructible for the stratification. But of course even being sufficiently careful this is remarkable (if true). Apr 8 at 22:06
• A trivial comment that addresses the stated question but does not address its emphasis: if you know Z but not X, you can Verdier specialize $\mathbb{Q}_X$ to the normal bundle of $Z$. This sheaf carries all the information of the nearby and vanishing cycles in the presence of the function $f$ (using it to trivialize the normal bundle).
– sdr
Apr 8 at 22:18
• @WillSawin: good point, in 2) I should have added "for any stratification for which the associated graded of $i^*\mathbb{Q}_X$ is constructible". However such a stratification is certainly "known" if I'm able to compute the weight filtration on $i^*\mathbb{Q}_X$. Apr 8 at 22:47
• If there was truly natural procedure for getting $Gr_W\psi_f(\mathbb{Q})$ from the central fibre, then it would presumably work in the category of mixed Hodge modules. But then it seems almost too much to expect, especially if $Z$ is Hodge-Tate type, and the nearby fibre isn't. I realize, I may be reading more into the question than what you asked, but I thought I'd point this out anyway. Apr 8 at 23:50
• A minor quibble: the map $f$ has to be proper for the cohomology of the nearby cycles to always agree with that of a nearby fibre.
– naf
Apr 13 at 8:18