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:
- 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?!
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 $\psi_f(\mathbb{Q}_X)$.)
