Bilipschitz invariants for germs of holomorphic foliations

TL;DR

This paper investigates bilipschitz equivalences of holomorphic foliation germs in complex two-space, establishing invariance of algebraic multiplicity and, for many cases, of the projective holonomy representation.

Contribution

It proves algebraic multiplicity is invariant under bilipschitz equivalences and identifies conditions under which the projective holonomy representation is also invariant.

Findings

Algebraic multiplicity is preserved under bilipschitz equivalences.

Projective holonomy representation is a bilipschitz invariant for many singularities.

Provides new invariants for classifying holomorphic foliation germs.

Abstract

In this paper we study bilipschitz equivalences of germs of holomorphic foliations in $(\mathbb{C}^2,0)$. We prove that the algebraic multiplicity of a singularity is invariant by such equivalences. Moreover, for a large class of singularities, we show that the projective holonomy representation is also a bilipschitz invariant.