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

TL;DR

This paper introduces an algorithm to compute the characteristic cycle of certain $\, ext{l}$-adic GKZ hypergeometric sheaves, linking algebraic and topological methods for understanding their geometric properties.

Contribution

It develops a new algorithm for calculating the characteristic cycle of $\, ext{l}$-adic GKZ hypergeometric sheaves and relates these sheaves to $\, ext{D}$-modules via the de Rham functor.

Findings

The algorithm successfully computes characteristic cycles for specific $\, ext{l}$-adic GKZ sheaves.

A topological model simplifies the determination of characteristic cycles.

The approach bridges algebraic and topological perspectives in sheaf theory.

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.