Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 0704.2995

Finite determination of regular (a,b)-modules

Daniel Barlet
View original source ↗Read source document ↗Export metadata ↓

Abstract

The concept of (a,b)-module comes from the study the Gauss-Manin lattices of an isolated singularity of a germ of an holomorphic function. It is a very simple ''abstract algebraic structure'', but very rich, whose prototype is the formal completion of the Brieskorn-module of an isolated singularity. The aim of this article is to prove a very basic theorem on regular (a,b)-modules showing that a given regular (a,b)-module is completely characterized by some ''finite order jet'' of its structure. Moreover a very simple bound for such a sufficient order is given in term of the rank and of two very simple invariants : the regularity order which count the number of times you need to apply \ $b^{-1}.a \simeq \partial_z.z$ in order to reach a simple pole (a,b)-module. The second invariant is the ''width'' which corresponds, in the simple pole case, to the maximal integral difference between to eigenvalues of $b^{-1}.a$ (the logarithm of the monodromy). In the computation of examples this theorem is quite helpfull because it tells you at which power of $b$ in the expansions you may stop without loosing any information.

Explore related subjects

math.CV

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Daniel Barlet. 2007-09-05. Finite determination of regular (a,b)-modules. https://arxiv.org/abs/0704.2995

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Ross-Witt Nyström correspondence and Ohsawa-Takegoshi extension

This paper is an attempt to study a few \emph{effective} universality phenomenons in complex geometry using deformations and the Hörmander $L^2$-method. The main idea of the proof is to deform the general case to the toric (or convex) case and to apply two new Berndtsson-Lempert type Ohsawa-Takegoshi extension theorems. The first one is based on the Ross-Witt Nyström correspondence picture, Darvas-Xia-Zhang's asymptotic slope formula for Ding-type functionals and Berndtsson's monotonicity theorem. The second version is proved using an $S^1$-symmetrization method for PSH potentials based on Witt Nyström's canonical Kähler deformations and Berndtsson-P\u aun's positivity theorem for Stein fibrations associated to deformations to normal bundles. The resulting effective universality phenomenons include a sharp lower bound of the Bergman kernel for compact Riemann surfaces, an effective Okounkov body construction and a sharp Faber-Widom type Bergman approximation of the logarithmic capacity.

math.CV↗
arXiv2026

Regularity of solutions to the complex Monge--Ampère equation with a general modulus of continuity

We investigate the regularity theory for the complex Monge--Ampère equation when, owing to the data or to the underlying domain, its solution fails to be in any Hölder class and to have a general concave modulus of continuity. Our first main result is motivated by, and generalizes, a result of S.-Y. Li. As an application, we prove a result on the regularity of solutions to the complex Monge--Ampère equation on general $B$-regular domains in $\mathbb{C}^n$.

math.CV↗
arXiv2026

Multiplicative Colombeau algebras and the Nyman--Beurling criterion for the Riemann hypothesis

This paper establishes an equivalence between the Riemann hypothesis and the association, together with uniform $L^2$-boundedness, of a moderate net in a Colombeau-type algebra built from polynomially damped Báez-Duarte sums. The regularization is performed by multiplicative (Mellin) convolution, which respects the dilation symmetry of the Beurling functions and guarantees that every approximant lies in the $L^2$-closure of the Beurling space. The equivalence is unconditional under the Riemann hypothesis: it uses only the qualitative convergence of Báez-Duarte, Mazur's theorem, and the classical Nyman--Beurling criterion. As a separate quantitative refinement, we prove that under two additional hypotheses on the non-trivial zeros of $ζ$ (simplicity and separation), the damping error admits a power-law bound with an explicit constant. This refinement is independent of the equivalence and is not used in its proof. The exponential damping $e^{-k\eps^2}$ is discussed as an open problem.

math.CV↗