Generalized Brieskorn Modules III. The algebra $\tilde{\mathcal{A}}\_{conv.}$

TL;DR

This paper introduces a convergent algebra acting on generalized Brieskorn modules linked to Gauss-Manin connections, extending formal results to a convergent setting and exploring their geometric and Hodge-theoretic implications.

Contribution

It generalizes previous formal results to a convergent algebra setting acting on Brieskorn modules, and connects these structures to geometric and Hodge-theoretic properties.

Findings

Defined the convergent algebra acting on holomorphic germs

Extended formal results to the convergent setting

Established links between algebraic structures and Hodge filtrations

Abstract

In this paper we introduce and study the ''convergent'' algebra (containing ''a'' and ''b'' and acting on holomorphic germs in ''a'') which naturally acts on the ''generalized Brieskorn modules'' associated to the Gauss-Manin connections of the germs at each point of the singular set of a holomorphic function on a complex manifold. We generalize to this convergent setting the results previously obtained (see 8], [9], [15] and [16]) in the formal case, and we show that, in suitable global situations (for instance when f is projective) we obtain also generalized (geometric) Brieskorn modules. So the question of the relationship between the left module structure on this algebra (which defines several interesting filtrations) and the mixte Hodge structure is raised