English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

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) whichcorresponds to the minimal filtered (regular) differential equation satisfiedby a relative de Rham cohomology class, began in [B.09] where the firststructure theorems are proved. Then in [B.10] we introduced the notion of themewhich corresponds in the \ $[\lambda]-$primitive case to frescos having aunique Jordan-H{\"o}lder sequence. Themes correspond to asymptotic expansion ofa given vanishing period, so to the image of a fresco in the module ofasymptotic expansions. For a fixed relative de Rham cohomology class (forinstance given by a smooth differential form $d-$closed and $df-$closed) eachchoice of a vanishing cycle in the spectral eigenspace of the monodromy for theeigenvalue \ $exp(2i\pi.\lambda)$ \ produces a \ $[\lambda]-$primitive theme,which is a quotient of the fresco associated to the given relative de Rhamclass itself. The first part of this paper shows that, for any \$[\lambda]-$primitive fresco there exists an unique Jordan-H{\"o}lder sequence(called the principal J-H. sequence) with corresponding quotients giving theopposite of the roots of the Bernstein polynomial in a non decreasing order.Then we introduce and study the semi-simple part of a given fresco and wecharacterize the semi-simplicity of a fresco by the fact for any given order ofthe roots of its Bernstein polynomial we may find a J-H. sequence making themappear with this order. Then, using the parameter associated to a rank \ 2 \ \$[\lambda]-$primitive theme, we introduce inductiveley a numerical invariant,that we call the \ $\alpha-$invariant, which depends polynomially on theisomorphism class of a fresco (in a sens which has to be defined) and whichallows to give an inductive way to produce a sub-quotient rank \ 2 \ theme of agiven \ $[\lambda]-$primitive fresco assuming non semi-simplicity. In the lastsection we prove a general existence result which naturally associate a frescoto any relative de Rham cohomology class of a proper holomorphic function of acomplex manifold onto a disc. This is, of course, the motivation for the studyof frescos.

Asymptotics of a vanishing period : General existence theorem and basic properties of frescos