Contruction of holomorphic parameters invariant by change of variable in the Gauss-Manin connection of an holomorphic map to a disc

TL;DR

This paper studies how holomorphic parameters related to (a,b)-modules and Gauss-Manin connections behave under changes of local coordinate near critical points, introducing invariants that remain stable under such transformations.

Contribution

It introduces two types of holomorphic parameters that are (quasi-)invariant under coordinate changes for frescos, advancing understanding of invariants in Gauss-Manin connections.

Findings

Proves stability of holomorphic families of frescos under coordinate change.

Constructs holomorphic parameters invariant or quasi-invariant under change of variable.

Provides tools to analyze invariants in the context of Gauss-Manin connections.

Abstract

When we consider a proper holomorphic map \ $\tilde{f}: X \to C$ \ of a complex manifold \ $X$ \ on a smooth complex curve \ $C$ \ with a critical value at a point \ $0$ \ in \ $C$, the choice of a local coordinate near this point allows to dispose of an holomorphic function \ $f$. Then we may construct, using this function, an (a,b)-modules structure on the cohomology sheaves of the formal completion (in \ $f$) \ of the complex of sheaves \ $(Ker\, df^{\bullet},d^{\bullet})$. These (a,b)-modules represent a filtered version of the Gauss-Manin connection of \ $f$. The most simple example of this construction is the Brieskorn module (see [Br.70]) of a function with an isolated singular point. See [B.08] for the case of a 1-dimensional critical locus. But it is clear that this construction depends seriously on the choice of the function \ $f$ \ that is to say on the choice of the local coordinate near the critical point \ $0$ \ in the complex curve \ $C$. The aim of the present paper is to study the behaviour of such constructions when we make a change of local coordinate near the origin. We consider the case of \ $[\lambda]-$primitive frescos, which are monogenic geometric (a,b)-modules corresponding to a minimal filtered differential equation associated to a relative de Rham cohomology class on \ $X$ \ (see [B.09-a] and [B.09-b]). An holomorphic parameter is a function on the set of isomorphism classes of frescos which behave holomorphically in an holomorphic family of frescos. In general, an holomorphic parameter is not invariant by a change of variable, but we prove a theorem of stability of holomorphic families of frescos by a change of variable and it implies that an holomorphic parameter gives again an holomorphic parameter by a change of variable. We construct here two different kinds of holomorphic parameters which are (quasi-)invariant by change of variable. The first kind is associated to Jordan blocks of the monodromy with size at least two. The second kind is associated to the semi-simple part of the monodromy and look like some "cross ratio" of eigenvectors. They allow, in the situation describe above, to associate to a given (vanishing) relative de Rham cohomology class some numbers, which will depend holomorphically of our data, and are independant of the choice of the local coordinate near \ $0$ \ to study the Gauss-Manin connection of this degeneration of compact complex manifolds.