Semicontinuity of the {\L}ojasiewicz exponent

TL;DR

This paper proves that the Łojasiewicz exponent, which measures the complexity of a holomorphic germ near a singularity, is lower semicontinuous under certain deformations, ensuring stability of this invariant.

Contribution

It establishes the lower semicontinuity of the Łojasiewicz exponent in multiplicity-constant deformations of holomorphic germs, a result previously unproven.

Findings

Łojasiewicz exponent is lower semicontinuous under deformations

Stability of the Łojasiewicz exponent in singularity theory

Advances understanding of invariants in complex analytic geometry

Abstract

We prove that the {\L}ojasiewicz exponent $l_0(f)$ of a finite holomorphic germ $f:(C^n,0)\to(C^n,0)$ is lower semicontinuous in any multiplicity-constant deformation of $f$.