Local Monodromy of Constructible Sheaves

Abstract

Given a morphism $f:X\to S$ of complex algebraic varieties and a constructible sheaf $\mathcal{G}$ on $X$, we compute the local monodromy of $R{f}_{\ast }(\mathcal{G})$ and $R{f}_{!}(\mathcal{G})$ 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.