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

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 \ $exp(2i\pi \mathrm{.}\lambda )$ \ produces a \ $[\lambda ]-$primitive theme, which is a quotient of the fresco associated to the given relative de Rham class 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 the opposite 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 we characterize the semi-simplicity of a fresco by the fact for any given order of the roots of its Bernstein polynomial we may find a J-H. sequence making them appear 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 the isomorphism class of a fresco (in a sens which has to be defined) and which allows to give an inductive way to produce a sub-quotient rank \ 2 \ theme of a given \ $[\lambda ]-$primitive fresco assuming non semi-simplicity. In the last section 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. This is, of course, the motivation for the study of frescos.