2011/01/20 by Daniel Barlet, Barlet, Daniel · 1 citation
Mathematics · #Advanced Topics in Algebra #Algebraic Geometry (math.AG) #Algebraic and Geometric Analysis #Complex Variables (math.CV) #FOS: Mathematics #Holomorphic and Operator Theory #math.AG #math.CV
paper · pdf · doi:10.48550/arxiv.1101.3959
43 pages
arxiv created 2011/01/20 · openalex publication_date 2011/01/20 · arxiv updated 2011/01/21 · 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. So the problem to determine which theme is a quotient of a given fresco is important to deduce possible asymptotic expansions of the various vanishing period integrals associated to a given relative de Rham class when we change the choice of the vanishing cycle. In the appendix 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.