On perturbations of singular complex analytic curves

TL;DR

This paper studies how small perturbations of singular complex analytic curves in a72 can still intersect the original curve, providing conditions under which intersection persists, with applications to higher dimensions and holomorphic multifunctions.

Contribution

It establishes a sufficient condition ensuring intersection persistence under small perturbations of complex analytic curves, extending to higher dimensions and holomorphic multifunctions.

Findings

Intersection persists for perturbations with at most one non-normal crossing discriminant point.

Provides a sufficient condition for intersection based on the discriminant point structure.

Extends results to higher-dimensional analogs and holomorphic multifunctions.

Abstract

Suppose $V$ is a singular complex analytic curve inside $\mathbb{C}^{2}$. We investigate when a singular or non-singular complex analytic curve $W$ inside $\mathbb{C}^{2}$ with sufficiently small Hausdorff distance $d_{H}(V, W)$ from $V$ must intersect $V$. We obtain a sufficient condition on $W$ which when satisfied gives an affirmative answer to our question. More precisely, we show the intersection is non-empty for any such $W$ that admits at most one non-normal crossing type discriminant point associated with some proper projection. As an application, we prove a special case of the higher-dimensional analog, and also a holomorphic multifunction analog of a result by Lyubich-Peters.