Bi-Lipschitz Invariants in Singularity Theory: Lojasiewicz Exponent and Euler Obstruction

TL;DR

This paper studies the bi-Lipschitz invariance of key local invariants in singularity theory, specifically the Łojasiewicz exponent and Euler obstruction, extending existing frameworks and addressing open questions.

Contribution

It extends the framework for invariance of these invariants to ideals in analytic function rings on affine toric varieties and answers an open question about the Euler obstruction's invariance.

Findings

The Łojasiewicz exponent is invariant under bi-Lipschitz maps.

The local Euler obstruction is invariant under bi-Lipschitz equivalence for certain hypersurfaces.

Conditions for invariance under non-degeneracy are established.

Abstract

In this work, we investigate the bi-Lipschitz invariance of two fundamental local invariants in singularity theory: the {\L}ojasiewicz exponent and the local Euler obstruction. We draw inspiration from Bivi\`a-Ausina and Fukui, whose framework we extend to ideals in rings of analytic functions defined on affine toric varieties. We establish conditions under which these invariants remain unchanged under bi-Lipschitz equivalence. We also provide an answer, to a particular case, to the open question of whether the local Euler obstruction is a bi-Lipschitz invariant. For hypersurfaces with isolated singularities, we show that the Euler obstruction is preserved under non-degeneracy conditions. These results contribute to the understanding of metric invariants in complex analytic geometry.