Deformations of abstract Brieskorn lattices

TL;DR

This paper investigates deformations of abstract Brieskorn lattices within Gauss-Manin systems, revealing finite-dimensional ambiguity and proving the non-existence of certain versal deformations, highlighting the importance of generation conditions.

Contribution

It demonstrates the finite-dimensional nature of deformation ambiguities and establishes the non-existence of versal deformations for Fourier transforms of abstract Brieskorn lattices.

Findings

Ambiguity in deformations is finite dimensional.

No versal deformation exists with expected dimension.

Generation condition is crucial for versal deformation existence.

Abstract

We study certain deformations of abstract Brieskorn lattices in fixed abstract Gauss-Manin systems, and show that the ambiguity of expressions of deformations coming from automorphisms of base spaces is essentially the same as the one coming from the choice of opposite filtrations, and hence is finite dimensional, although the freedom of parameters in the expressions of deformations is infinite dimensional. As a consequence, we can prove the non-existence of a versal deformation of the Fourier transform of this abstract Brieskorn lattice with expected dimension. This shows that the generation condition is quite essential for the existence of versal deformations with expected dimensions in the absolute case.