The characteristic cycle of a non-confluent $\ell$-adic GKZ hypergeometric sheaf

Abstract

An $\ell$-adic GKZ hypergeometric sheaf is defined analogously to a GKZ hypergeometric $\mathcal{D}$-module. We introduce an algorithm of computing the characteristic cycle of an $\ell$-adic GKZ hypergeometric sheaf of certain type. Our strategy is to apply a formula of the characteristic cycle of the direct image of an $\ell$-adic sheaf. We verify the requirements for the formula to hold by calculating the dimension of the direct image of a certain closed conical subset of cotangent bundle. We also define an $\ell$-adic GKZ-type sheaf whose specialization tensored with a constant sheaf is isomorphic to an $\ell$-adic non-confluent GKZ hypergeometric sheaf. On the other hand, the topological model of an $\ell$-adic GKZ-type sheaf is isomorphic to the image by the de Rham functor of a non-confluent GKZ hypergeometric $\mathcal{D}$-module whose characteristic cycle has been calculated. This gives an easier way to determine the characteristic cycle of an $\ell$-adic non-confluent GKZ hypergeometric sheaf of certain type.