Asymptotics of a vanishing period : characterization of semi-simplicity

Abstract

In this paper we introduce the word {\em fresco} to denote a monogenic geometric (a,b)-module. This "basic object" (generalized Brieskorn module with one generator) corresponds to the formal germ of the minimal filtered (regular) differential equation. Such an equation is satisfied by a relative de Rham cohomology class at a critical value of a holomorphic function on a smooth complex manifold. In [B.09] the first structure theorems are proved. Then in [B.10] we introduced the notion of {\em theme} which corresponds in the \ $[\lambda]-$primitive case to frescos having a unique Jordan-H{\"o}lder sequence (a unique Jordan block for the monodromy). Themes correspond to asymptotic expansion of a given vanishing period, so to an 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. \\ We show that for any fresco there exists an {\em unique} Jordan-H{\"o}lder sequence, called the {\em principal J-H. sequence}, with corresponding quotients giving the opposite of the roots of the Bernstein polynomial in increasing order. We 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 on the roots of its Bernstein polynomial we may find a J-H. sequence making them appear with this order. Then we construct a numerical invariant, called the \ $\beta-$invariant, and we show that it produces numerical criteria in order to give a necessary and sufficient condition on a fresco to be semi-simple. We show that these numerical invariants define a natural algebraic stratification on the set of isomorphism classes of fresco with given fundamental invariants (or equivalently with given roots of the Bernstein polynomial).