Contruction of quasi-invariant holomorphic parameters for the Gauss-Manin connection of a holomorphic map to a curve (second version)

TL;DR

This paper develops holomorphic parameters for frescos, filtered differential equations with regular singularities, that remain quasi-invariant under variable changes, aiding the study of degenerations in complex geometry.

Contribution

It constructs a locally versal family of frescos for each Bernstein polynomial and introduces quasi-invariant holomorphic parameters under variable changes.

Findings

Constructed a locally versal holomorphic family for fixed Bernstein polynomial.

Developed holomorphic parameters that are quasi-invariant under variable changes.

Applied to relative de Rham cohomology in degenerations of complex manifolds.

Abstract

In this paper we consider holomorphic families of frescos (i.e. filtered differential equations with a regular singularity) and we construct a locally versal holomorphic family for every fixed Bernstein polynomial. We construct also several holomorphic parameters (a holomorphic parameter is a function defined on a set of isomorphism classes of frescos) which are quasi-invariant by changes of variable. This is motivated by the fact that a fresco is associated to a relative de Rham cohomology class on a one parameter degeneration of compact complex manifolds, up to a change of variable in the parameter. Then the value of a quasi-invariant holomorphic parameter on such data produces a holomorphic (quasi-)invariant of such a situation.