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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. [Edit 2023: Chris Hone explained an elementary argument that shows that this is indeed the case, so this shouldn't have been surprising to me. I'm happy to reproduce the argument if anyone is interested.]
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 $\psi_f(\mathbb{Q}_X)$.)

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

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

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. [Edit 2023: Chris Hone explained an elementary argument that shows that this is indeed the case, so this shouldn't have been surprising to me. I'm happy to reproduce the argument if anyone is interested.]
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 $\psi_f(\mathbb{Q}_X)$.)

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

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