On the comparison of nearby cycles via $b$-functions

TL;DR

This paper provides a simplified proof of the comparison between nearby and vanishing cycles within the Riemann-Hilbert correspondence framework, avoiding the complex Kashiwara-Malgrange $V$-filtrations.

Contribution

It introduces a new, streamlined proof method inspired by Beilinson and Bernstein, bypassing traditional $V$-filtration techniques.

Findings

Simplified proof of nearby and vanishing cycles comparison

Avoidance of Kashiwara-Malgrange $V$-filtrations

Enhanced understanding of Riemann-Hilbert correspondence

Abstract

In this article, we give a simple proof of the comparison of nearby and vanishing cycles in the sense of Riemann-Hilbert correspondence following the idea of Beilinson and Bernstein, without using the Kashiwara-Malgrange $V$-filtrations.