Gauss-Manin Connections for Boundary Singularities and Isochore Deformations

TL;DR

This paper extends classical theorems on Gauss-Manin connections to boundary singularities, exploring their properties, monodromy, and asymptotics, with applications to isochore deformation theory and volume-preserving transformations.

Contribution

It introduces relative analogs of key theorems for boundary singularities, linking Gauss-Manin connections to isochore deformations and volume form normal forms.

Findings

Proves relative versions of classical theorems for boundary singularities.

Establishes relations between Gauss-Manin connections, monodromy, and asymptotics.

Provides applications to isochore deformation theory and volume-preserving diffeomorphisms.

Abstract

We study here the relative cohomology and the Gauss-Manin connections associated to an isolated singularity of a function on a manifold with boundary, i.e. with a fixed hyperplane section. We prove several relative analogs of classical theorems obtained mainly by E. Brieskorn and B. Malgrange, concerning the properties of the Gauss-Manin connection as well as its relations with the Picard-Lefschetz monodromy and the asymptotics of integrals of holomorphic forms along the vanishing cycles. Finally, we give an application in isochore deformation theory, i.e. the deformation theory of boundary singularities with respect to a volume form. In particular we prove the relative analog of J. Vey's isochore Morse lemma, J. -P. Fran\c{c}oise's generalisation on the local normal forms of volume forms with respect to the boundary singularity-preserving diffeomorphisms, as well as M. D. Garay's theorem on the isochore version of Mather's versal unfolding theorem.