Asymptotics of a vanishing period : the quotient themes of a given fresco

TL;DR

This paper introduces the concept of 'fresco' as a fundamental geometric module related to differential equations and asymptotic expansions of vanishing periods, exploring their structure and quotients.

Contribution

It defines 'fresco' and 'theme' modules, studies their properties, and establishes conditions for themes to be quotients of frescos, advancing understanding of asymptotic behaviors in complex geometry.

Findings

Defined 'fresco' as a primitive monogenic geometric (a,b)-module.

Proved existence of frescos associated with relative de Rham cohomology classes.

Characterized when a theme is a quotient of a fresco.

Abstract

In this paper we introduce the word "fresco" to denote a $[\lambda]-$primitive monogenic geometric (a,b)-module. The study of this "basic object" (generalized Brieskorn module with one generator) which corresponds to the minimal filtered (regular) differential equation satisfied by a relative de Rham cohomology class, began in [B.09] where the first structure theorems are proved. Then in [B.10] we introduced the notion of theme which corresponds in the $[\lambda]-$primitive case to frescos having a unique Jordan-H{\"o}lder sequence. Themes correspond to asymptotic expansion of a given vanishing period, so to the image of a fresco in the module of asymptotic expansions. For a fixed relative de Rham cohomology class (for instance given by a smooth differential form $d-$closed and $df-$closed) each choice of a vanishing cycle in the spectral eigenspace of the monodromy for the eigenvalue $exp(-2i\pi.\lambda)$ produces a $[\lambda]-$primitive theme, which is a quotient of the fresco associated to the given relative de Rham class itself. So the problem to determine which theme is a quotient of a given fresco is important to deduce possible asymptotic expansions of the various vanishing period integrals associated to a given relative de Rham class when we change the choice of the vanishing cycle. In the appendix we prove a general existence result which naturally associate a fresco to any relative de Rham cohomology class of a proper holomorphic function of a complex manifold onto a disc.