Hermitian $(a, b)$-modules and Saito's "higher residue pairings"

TL;DR

This paper investigates the existence and structure of Hermitian forms on regular (a, b)-modules, extending previous work to include higher residue pairings and providing explicit examples and applications.

Contribution

It proves that regular (a, b)-modules with non-degenerate bilinear forms decompose uniquely into modules with Hermitian or anti-Hermitian forms, and extends these results to modules associated with isolated singularities.

Findings

Every regular (a, b)-module with a non-degenerate bilinear form decomposes uniquely.

Explicit examples of modules with Hermitian and anti-Hermitian forms are provided.

The results connect to Saito's higher residue pairings, confirming their axioms.

Abstract

Following the work of Daniel Barlet ([Bar97]) and Ridha Belgrade ([Bel01]) the aim of this article is the study of the existence of $(a, b)$-hermitian forms on regular $(a, b)$-modules. We show that every regular $(a,b)$-module with a non-degenerate bilinear form can be written in an unique way as a direct sum of $(a, b)$-modules $E_i$ that admit either an $(a, b)$-hermitian or an $(a, b)$-anti-hermitian form or both; all three cases are equally possible with explicit examples. As an application we extend the result in [Bel01] on the existence for all $(a, b)$-modules $E$ associated with the Brieskorn module of a holomorphic function with an isolated singularity, of an $(a,b)$-bilinear non degenerate form on $E$. We show that with a small transformation Belgrade's form can be considered $(a, b)$-hermitian and that the result satis es the axioms of Kyoji Saito's "higher residue pairings".