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

Volume 108 / 2013

Piotr P. Karwasz Annales Polonici Mathematici 108 (2013), 241-261 MSC: Primary 32S25; Secondary 32S40, 32S50. DOI: 10.4064/ap108-3-4


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”.


  • Piotr P. KarwaszInstytut Matematyki
    Uniwersytet Gdański
    80-952 Gdańsk, Poland

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