Local Monodromy of Constructible Sheaves
Madhav V. Nori, Deepam Patel
TL;DR
This work develops a comprehensive framework for the local monodromy of pushforwards of constructible sheaves under morphisms between complex algebraic varieties. By introducing and exploiting boundary spectra ${\rm BSp}$ and the refined reduced spectra ${\rm Sp}_{red}$ associated to analytic loops, the authors establish uniform, root-controlled bounds for the spectra of $R^qf_*(\mathcal{G})$ and $R^qf_!(\mathcal{G})$, with sharp statements when $\dim S=1$ and in general. The results generalize Grothendieck’s quasi-unipotence theorem to sheaves of $R$-modules and to non-geometric local systems, and they yield immediate applications to integral transforms, intersection cohomology, and the monodromy of abelian covers and Alexander modules. The paper also develops a robust technical apparatus—nearby cycles, boundary monoids, and group-cohomology tools—that enable uniform control of monodromy across families and explicit computations in abelian-cover settings, with significant implications for torsion coefficients and generalized Alexander theory.
Abstract
Given a morphism $f: X \rightarrow S$ of complex algebraic varieties and a constructible sheaf $\mathcal{G}$ on $X$, we compute the local monodromy of $Rf_*(\mathcal{G})$ and $Rf_!(\mathcal{G})$ in terms of the local monodromy of $\mathcal{G}$. Our results generalize previous results by Brieskorn, Borel, Clemens, Deligne, Landsman, Griffiths, Grothendieck, and Kashiwara in the setting of quasi-unipotent sheaves. In the following, we consider the general setting of sheaves of $R$-modules for a commutative noetherian ring $R$, and give applications to computing local monodromy of abelian covers in a uniform manner. We also obtain applications in the context of `generalized Alexander modules' and intersection cohomology with torsion coefficients.