消失周期的渐近性：半单性的刻画
Asymptotics of a vanishing period : characterization of semi-simplicity
摘要：
本文引入了词{ \em fresco}来表示单演几何（a，b）-模.这个“基本对象”（广义Brieskorn模块与一个发电机）对应的最小过滤（定期）微分方程的形式胚芽。在光滑复流形上的一个全纯函数的临界值处，相对de Rham上同调类满足这样一个方程。在[B.09]中证明了第一结构定理。然后在[B.10]中，我们引入了{ \em主题}的概念，它在\ $[λ]-$原始情况下对应于具有唯一约旦-H {<$}lder序列（单值性的唯一约旦块）的壁画。主题对应于给定消失周期的渐近展开，因此对应于渐近展开模块中的壁画图像。对于一个固定的相对德·拉姆上同调类（例如由光滑微分形式$d-$闭和$df-$闭给出），在特征值\ $exp(2iπ.λ)$ \的单值谱特征空间中每选择一个消失环都会产生一个\ $[λ]-$本原主元，它是与给定的相对德·拉姆类本身相关联的fresco的商。我们证明了对于任何壁画都存在一个{ \em unique} Jordan-H{<$}lder序列，称为{ \em principal J-H.序列}，相应的导数给出了伯恩斯坦多项式根的相反方向。我们研究了一幅壁画的半单部分，我们通过对它的伯恩斯坦多项式的根的任何给定阶，我们可以找到一个J-H来刻画壁画的半单性。使它们按这个顺序出现。然后，我们构建了一个数值不变量，称为\ $β-$不变，我们表明，它产生的数值标准，以给出一个壁画是半简单的充要条件。我们发现，这些数值不变量定义一个自然的代数分层上的壁画的同构类与给定的基本不变量（或等价于给定的根的伯恩斯坦多项式）。
英文摘要：
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 \ $[λ]-$primitive case to frescos having a unique Jordan-H{ö}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 \ $exp(2iπ.λ)$ \ produces a \ $[λ]-$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{ö}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 \ $β-$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).
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