Riemann-Hilbert correspondence for Alexander complexes

TL;DR

This paper develops a relative Riemann-Hilbert correspondence for Alexander complexes using equivariant regular holonomic D-modules, providing a global perspective and a formula for their support via Bernstein-Sato ideals.

Contribution

It introduces a novel relative Riemann-Hilbert correspondence for Alexander complexes employing equivariant D-modules, extending the understanding of Deligne's nearby cycles.

Findings

Established a global Riemann-Hilbert correspondence for Alexander complexes.

Derived a formula for the relative support of Alexander complexes using Bernstein-Sato ideals.

Provided a new approach to the study of nearby cycles in algebraic geometry.

Abstract

We establish a relative Riemann-Hilbert correspondence for Alexander complexes (also known as Sabbah specialization complexes) by using relative regular holonomic $\mathscr D$-modules in an equivariant way, which particularly gives a "global" approach to the correspondence for Deligne's nearby cycles. Using the correspondence and zero loci of Bernstein-Sato ideals, we obtain a formula for the relative support of the Alexander complexes.