Two finiteness theorem for $(a,b)$-module

TL;DR

This paper establishes two finiteness theorems for (a,b)-modules, one constructing geometric modules from proper holomorphic functions and another showing that regular (a,b)-modules are determined by finite truncations.

Contribution

It introduces new finiteness results for (a,b)-modules, linking geometric modules to global functions and providing explicit bounds for module classification.

Findings

Constructed geometric (a,b)-modules from proper holomorphic functions.

Proved regular (a,b)-modules are determined by finite truncations.

Demonstrated (a,b)-modules can be classified using explicit invariants.

Abstract

We prove the following two results 1. For a proper holomorphic function $ f : X \to D$ of a complex manifold $X$ on a disc such that $\{df = 0 \} \subset f^{-1}(0)$, we construct, in a functorial way, for each integer $p$, a geometric (a,b)-module $E^p$ \ associated to the (filtered) Gauss-Manin connexion of $f$. This first theorem is an existence/finiteness result which shows that geometric (a,b)-modules may be used in global situations. 2. For any regular (a,b)-module $E$ we give an integer $N(E)$, explicitely given from simple invariants of $E$, such that the isomorphism class of $E\big/b^{N(E)}.E$ determines the isomorphism class of $E$. This second result allows to cut asymptotic expansions (in powers of $b$) \ of elements of $E$ without loosing any information.