2012/01/13 by Daniel Barlet, Barlet, Daniel · 1 citation
Mathematics · #Advanced Topics in Algebra #Algebraic Geometry (math.AG) #Algebraic and Geometric Analysis #Algebraic structures and combinatorial models #Complex Variables (math.CV) #FOS: Mathematics #math.AG #math.CV
paper · pdf · doi:10.48550/arxiv.1201.2757
38 pages. arXiv admin note: substantial text overlap with arXiv:1101.3959
arxiv created 2012/01/13 · openalex publication_date 2012/01/13 · arxiv updated 2012/01/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper we introduce the word "fresco" to denote a [λ]-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 [λ]-primitive case to frescos having a unique Jordan-Hölder 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(2iπ.λ) produces a [λ]-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 [λ]-primitive fresco there exists an unique Jordan-Hölder 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 [λ]-primitive theme, we introduce inductiveley a numerical invariant, that we call the α-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 [λ]-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.