Hermitian $(a,b)$-modules and Saito's “higher residue pairings”
Following the work of Daniel Barlet [Pitman Res. Notes Math. Ser. 366 (1997), 19–59] and Ridha Belgrade [J. Algebra 245 (2001), 193–224], the aim of this article is to study the existence of $(a,b)$-hermitian forms on regular $(a,b)$-modules. We show that every regular $(a,b)$-module $E$ with a non-degenerate bilinear form can be written in a 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 possible, and we give explicit examples.
As an application we extend the result of Ridha Belgrade 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 satisfies the axioms of Kyoji Saito's “higher residue pairings”.