Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On perturbations of singular complex analytic curves

Achinta Kumar Nandi

TL;DR

The paper addresses when perturbations of a singular 1-dimensional complex analytic curve necessarily intersect the original curve, focusing on small Hausdorff perturbations near a singular point. It develops a concrete sufficient condition: in a good neighborhood, a nearby 1-dimensional variety $W$ with at most one non-normal crossing discriminant point relative to a projection has to intersect $V$ for sufficiently small $d_H(V,W)$, with the threshold depending on the neighborhood. The approach combines Puiseux parametrizations, discriminant-stability under perturbations, and Lyubich-Peters-type intersection arguments, and extends to holomorphic multifunctions and a higher-dimensional analog for finite holomorphic mappings. Together, these results illuminate the local stability of singular curves under perturbations and connect complex-analytic geometry with dynamical considerations and finite-mapping theory.

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.

On perturbations of singular complex analytic curves

TL;DR

The paper addresses when perturbations of a singular 1-dimensional complex analytic curve necessarily intersect the original curve, focusing on small Hausdorff perturbations near a singular point. It develops a concrete sufficient condition: in a good neighborhood, a nearby 1-dimensional variety with at most one non-normal crossing discriminant point relative to a projection has to intersect for sufficiently small , with the threshold depending on the neighborhood. The approach combines Puiseux parametrizations, discriminant-stability under perturbations, and Lyubich-Peters-type intersection arguments, and extends to holomorphic multifunctions and a higher-dimensional analog for finite holomorphic mappings. Together, these results illuminate the local stability of singular curves under perturbations and connect complex-analytic geometry with dynamical considerations and finite-mapping theory.

Abstract

Suppose is a singular complex analytic curve inside . We investigate when a singular or non-singular complex analytic curve inside with sufficiently small Hausdorff distance from must intersect . We obtain a sufficient condition on which when satisfied gives an affirmative answer to our question. More precisely, we show the intersection is non-empty for any such 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.
Paper Structure (17 sections, 6 theorems, 38 equations)

This paper contains 17 sections, 6 theorems, 38 equations.

Table of Contents

  1. Introduction
  2. Notational Conventions
  3. Acknowledgements
  4. Background
  5. Good Neighborhoods, Proper Projections, and their Discriminant Sets
  6. Holomorphic Multifunctions
  7. Hausdorff Distance
  8. The Puiseux Parametrization Theorem
  9. A Sufficient Condition
  10. Separation of branches away from the discriminant set
  11. Observations concerning the discriminant set
  12. On the maximal number of distinct branches
  13. Proof of the Main Result
  14. Applications of the sufficient condition
  15. A holomorphic multifunction analog
  16. ...and 2 more sections

Key Result

Lemma 3.1

Suppose $V$ is an $r$-dimensional variety, $p \in V$, $H^{n} = H^{r} \times \Delta^{'(n-r)}$ is a neighborhood of $p$ that is good for $V$, and $D_V \subset H^{r}$ is the associated discriminant set. Further suppose $\Delta^{r} \subset H^{r}$ is a pre-compact neighborhood such that $p$ is in $\Delta

Theorems & Definitions (19)

  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • Lemma 3.4
  • proof
  • Lemma 3.5
  • proof
  • proof
  • ...and 9 more