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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 $\psi_f(\mathbb{Q}_X)$.)

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    $\begingroup$ 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). $\endgroup$
    – Will Sawin
    Apr 8 at 22:06
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    $\begingroup$ 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). $\endgroup$
    – sdr
    Apr 8 at 22:18
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    $\begingroup$ @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 $i^*\mathbb{Q}_X$. $\endgroup$ Apr 8 at 22:47
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    $\begingroup$ 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. $\endgroup$ Apr 8 at 23:50
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    $\begingroup$ 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. $\endgroup$
    – naf
    Apr 13 at 8:18

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