A Deligne-Malgrange Riemann-Hilbert correspondence for closed 1-forms

TL;DR

This paper develops a Deligne-Malgrange Riemann-Hilbert correspondence for closed 1-forms, inspired by Kontsevich-Soibelman's conjecture, and applies it to compare algebraic 1-forms on complex curves.

Contribution

It introduces a new Riemann-Hilbert correspondence for closed 1-forms and proves a related isomorphism comparison theorem for algebraic 1-forms on complex curves.

Findings

Established a Deligne-Malgrange type Riemann-Hilbert correspondence.

Proved a variant of the isomorphism comparison theorem for algebraic 1-forms.

Connected the correspondence to Kontsevich-Soibelman's conjecture.

Abstract

Motivated by the work of Kontsevich-Soibelman on the comparison of isomorphisms conjecture for closed algebraic $1$-forms, we establish a Riemann-Hilbert correspondence of Deligne-Malgrange type. As an application, we prove a variant of the comparison of isomorphisms theorem for a simple class of algebraic $1$-forms on complex curves.