Iterating specialization of sheaves?

Question

This is a question about the operation of taking the specialization of sheaves along a subspace. I'll recall the settings in which I've encountered a notion of specialization of sheaves:

• The real, differential setting: Given a manifold $X$ and a submanifold $M$, we may form the deformation of $X$ to the normal bundle of $M$, which I'll denoted by $Df_M(X)$. The deformation to the normal bundle comes equipped with two maps: $t: Df_M(X) \to \mathbb{R}$ and $p: Df_M(X) \to X$. Here's a list of maps we'll need to define the specialization: $j: t^{-1}(\mathbb{R}_{>0}) \hookrightarrow Df_M(X)$, $i: N_{M/X} \simeq t^{-1}(0) \hookrightarrow Df_M(X)$, and $\tilde{p} := p|_{t^{-1}(\mathbb{R}_{>0})}$. Given a sheaf $F$, we define the specialization of $F$ along $M$ as follows: $\nu_M(F) := i^*j_*\tilde{p}^*F$. This construction is laid out in great detail in chapter four of Sheaves on Manifolds by Kashiwara, Schapira.
• The algebraic setting wherever we have a notion of nearby cycles: Given a finite type $k$-scheme $X$ and a subscheme $Z$, we may again form the deformation of $X$ to the normal cone of $Z$, which we again denote by $Df_Z(X)$. This construction is given affine locally by the Rees algebra of the ideal defining $Z$, and it again comes equipped with maps $t: Df_Z(X) \to \mathbb{A}^1$ and $p: Df_Z(X) \to X$. Here are two maps similar to above: $j: Df_Z(X) \times_{\mathbb{A}^1} \mathbb{A}^{1*} := Df_Z(X)_{\mathbb{A}^{1*}} \hookrightarrow DF_Z(X)$ and $\tilde{p} = p|_{Df_Z(X)_{\mathbb{A}^{1*}}}$. Then the specialization functor is defined as $\nu_Z(F) := \psi_t(j_!\tilde{p}^*F)$ or equivalently $\psi_t(j_*\tilde{p}^*F)$, where $\psi_t$ is nearby cycles with respect to $t$. This construction is detailed in section 8 of this paper by Verdier.

There is probably also a D-module setting, but I don't know about this.

I'm primarily interested in the algebraic setting for a scheme over $\mathbb{C}$, in which I take the analytification of $Df_Z(X)$ and perform the usual topological nearby cycles for an analytic map to the complex disk.

Question: Suppose I have two closed subschemes of $X$, $Z_1$ and $Z_2$, such that $Z_2$ is a closed subscheme of $Z_1$ and such that both have well-defined normal bundles in $X$ (e.g. both are l.c.i). When is it true that $\nu_{Z_2}(F) \simeq \nu_{Z_2}(\nu_{Z_1}(F))$, where on the right-hand side, $Z_2$ is viewed as a subset of the zero section of the normal bundle $N_{Z_1/X}$?

I'll feel embarrassed if the answer to this question is something simple like "use a base-change," but at first glance it looks to me like the question might have some connection to the problem of commuting nearby cycles or nearby cycles over higher dimensional bases, which are hard problems. I would also appreciate an answer to this question in any of the other settings above, replacing closed subscheme by closed submanifold in the real differential setting, and analytic sheaves with étale sheaves in the other algebraic settings. Any vague intuitions or tangential insights are also welcome.

Answer 1

Iterating specialization is likely as hard as anything else (I would not be surprised to learn you can encode any computer program into a question of whether $\nu_X \nu_Y F = \nu_X F$ for some $X \subset Y$ and some $F$.)

There is a way to give microlocal criteria for when such an equality holds (which applies similarly to commuting nearby cycles, etc). You can express some stalk or local space of sections of say $\nu_X F$ in terms of some limit over some diagram of open sets in the original space, let us say $U_\alpha$. Similarly stalks of $\nu_X \nu_Y F$ will have a similar description in terms of some $V_\beta$. Now if you can identify $F(U_\alpha)$ and $F(V_\beta)$ for appropriate cofinal sequences of $\alpha, \beta$, you will be done. So you study how to deform $U_\alpha$ to $V_\beta$, and check that this is noncharacteristic for $F$.

You can find some examples of such arguments in a related setting in section 4 of this article https://arxiv.org/pdf/2007.10154.pdf