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Apologies if this belongs on MSE, but none of the tags I wanted existed, so I took it as a hint to post on MO.

Edit: here is my definition of nearby cycles. Suppose $X$ is a complex analytic space with a map $f: X \to \mathbb{A}^1$, and $F$ is a sheaf on $X$. Then $\psi_f(F) := i^*j_*\pi_*\pi^*j^*F$ where: $i$ is the inclusion of the zero fiber, $j$ is the inclusion of its complement $X^*$, and $\pi: \tilde{X^*} \to X^*$, where $\tilde{X^*}$ is the product over $\mathbb{A}^{1*}$ of its universal cover and $X^*$.

For simplicity, let's suppose I have a complex rank one local system $\mathscr{L}_{\lambda}$ on $\mathbb{G}_m$ with monodromy given by multiplication with complex number $\lambda$. The nearby cycles $\psi_{id}(\mathscr{L}_{\lambda})$ for identity map $id: \mathbb{G}_m \to \mathbb{G}_m$ is the vector space $\mathbb{C}$ considered as a stalk at $0 \in \mathbb{A}^1$. It has a monodromy operator $M: \psi_{id}(\mathscr{L}_{\lambda}) \to \psi_{id}(\mathscr{L}_{\lambda})$ like all nearby cycles, and it's common knowledge that this operator is multiplication by $\lambda$. But why?

I'd like an explanation that doesn't appeal to Milnor fibers. Looking purely at the definition of the nearby cycles functor, it seems to me like $\psi_{id}(\mathscr{L}_{\lambda})$ should have trivial monodromy operator because the pullback of such a local system along the universal cover of $\mathbb{G}_m$ is the constant sheaf with stalk $\mathbb{C}$. I am clearly confused about something, but not sure what...

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  • $\begingroup$ Which definition, specifically, do you want to use? $\endgroup$
    – Will Sawin
    Commented Aug 19, 2021 at 2:45
  • $\begingroup$ I'll edit to add precise definition to question, but the topological one--so not the etale one in particular. $\endgroup$
    – Mathmank
    Commented Aug 19, 2021 at 2:58
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    $\begingroup$ So the monodromy operator is going to arise from an automorphism of the universal cover of $\mathbb A^{1*}$, which gives an automorphism of $\tilde{X}^*$, thus an automorphism of $\pi_* \pi^*$. Similarly the global monodromy of $\mathbb G_m$ is going to arise from the action of an automorphism of the universal cover of $\mathbb G_m$. Then you will use the fact that these are the same universal cover and the same automorphism. $\endgroup$
    – Will Sawin
    Commented Aug 19, 2021 at 3:15
  • $\begingroup$ Maybe that's what I don't understand: you're saying the monodromy operator of my local system arises from an automorphism of $exp_*exp^*$ ($exp$ here being the universal cover of $\mathbb{G}_m$)? Applied to what sheaf? $\endgroup$
    – Mathmank
    Commented Aug 19, 2021 at 4:56
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    $\begingroup$ The monodromy of any local system on any variety is equivalent to the action of the automorphisms of the universal cover on the pullback of that local system to the universal cover. Note that the pullback of the local system to the universal cover is the constant sheaf, so automorphisms give a well-defined action on av ector space, and that the automorphism group of the universal cover over the base is the fundamental group (deck transformations). $\endgroup$
    – Will Sawin
    Commented Aug 19, 2021 at 12:37

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