Invariants of Relative Right and Contact Equivalences

TL;DR

This paper extends classical invariants like the Milnor and Tjurina algebras to classify holomorphic function germs relative to an analytic variety, providing new criteria for equivalence under relative right and contact transformations.

Contribution

It establishes that isomorphic relative Milnor algebras imply relative right equivalence for quasihomogeneous polynomials, and shows the relative Tjurina algebra is a complete invariant under certain conditions.

Findings

Relative Milnor algebra determines relative right equivalence.

Relative Tjurina algebra classifies function germs under relative contact equivalence.

Results generalize classical invariants to the relative setting.

Abstract

We study holomorphic function germs under equivalence relations that preserve an analytic variety. We show that two quasihomogeneous polynomials, not necessarily with isolated singularities, having isomorphic relative Milnor algebras are relative right equivalent. Under the condition that the module of vector fields tangent to the variety is finitely generated, we also show that the relative Tjurina algebra is a complete invariant for the classification of arbitrary function germs with respect to the relative contact equivalence. This is the relative version of a well known result by Mather and Yau.