Holomorphic families of $[\lambda]-$primitive themes

TL;DR

This paper develops a framework for classifying and deforming $[\lambda]-$primitive themes using finite parameters, constructing universal families in many cases and providing examples where universality fails.

Contribution

It introduces a locally versal holomorphic deformation theory for $[\lambda]-$primitive themes based on Bernstein polynomials, extending previous classification results.

Findings

Constructed locally versal holomorphic deformations for $[\lambda]-$primitive themes.

Proved universality of canonical families in many cases.

Provided examples where no local universal family exists.

Abstract

This article is the continuation of [B. 13-b] where we show how the isomorphism class of a $[\lambda]-$primitive theme with a given Bernstein polynomial may be characterized by a (small) finite number of complex parameters. We construct here a corresponding locally versal holomorphic deformation of $ [\lambda]-$primitive themes for each given Bernstein polynomial. Then we prove the universality of the corresponding "canonical family" in many cases. We also give some examples where no local universal family exists.