Natural Commuting of Vanishing Cycles and the Verdier Dual

TL;DR

This paper proves that shifted vanishing and nearby cycles commute with Verdier duality via a natural isomorphism, extending previous results to coefficients beyond fields.

Contribution

It establishes the commutation of vanishing and nearby cycles with Verdier duality in a more general setting with arbitrary coefficients.

Findings

Vanishing cycles commute with Verdier duality up to a natural isomorphism.

The result holds even when coefficients are not in a field.

Extends known commutation results to broader coefficient systems.

Abstract

We prove that the shifted vanishing cycles and nearby cycles commute with Verdier dualizing up to a {\bf natural} isomorphism, even when the coefficients are not in a field.