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

Piotr P. Karwasz¹

¹ Instytut Matematyki Uniwersytet Gdański 80-952 Gdańsk, Poland

Annales Polonici Mathematici, Tome 108 (2013) no. 3, pp. 241-261.

Résumé

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”.

Détail

DOI : 10.4064/ap108-3-4

Keywords: following work daniel barlet pitman res notes math ser ridha belgrade algebra article study existence hermitian forms regular modules every regular module non degenerate bilinear form written unique direct sum modules admit either hermitian anti hermitian form three cases possible explicit examples application extend result ridha belgrade existence modules associated brieskorn module holomorphic function isolated singularity bilinear non degenerate form small transformation belgrades form considered hermitian result satisfies axioms kyoji saitos higher residue pairings