Local Monodromy of Constructible Sheaves

TL;DR

This paper develops a method to compute the local monodromy of derived pushforwards of constructible sheaves in complex algebraic geometry, generalizing classical results and applying to abelian covers and intersection cohomology.

Contribution

It generalizes previous monodromy computations to sheaves of R-modules and provides a unified approach with applications to abelian covers and intersection cohomology.

Findings

Computed local monodromy of derived pushforwards in general sheaf settings

Unified approach to monodromy of abelian covers

Applications to generalized Alexander modules and intersection cohomology

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.