A Converse to a Theorem on Normal Forms of Volume Forms with Respect to a Hypersurface

TL;DR

This paper proves a converse to a theorem by Varchenko, showing that two germs of volume forms are equivalent under hypersurface-preserving diffeomorphisms if their difference is an exact form on the smooth part of the hypersurface.

Contribution

It provides a positive answer to a question by Y. Colin de Verdière, establishing a converse to Varchenko's theorem on volume forms and hypersurface singularities.

Findings

Confirmed the converse of Varchenko's theorem.

Established conditions for equivalence of volume forms under hypersurface-preserving diffeomorphisms.

Extended understanding of the structure of volume forms near hypersurface singularities.

Abstract

In this note we give a positive answer to a question asked by Y. Colin de Verdi\`ere concerning the converse of the following theorem, due to A. N. Varchenko: two germs of volume forms are equivalent with respect to diffeomorphisms preserving a germ of an isolated hypersurface singularity, if their difference is the differential of a form whose restriction on the smooth part of the hypersurface is exact.