On perturbations of singular complex analytic curves
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arXiv
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866912221525704704 |
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| author | Nandi, Achinta Kumar |
| author_facet | Nandi, Achinta Kumar |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_11656 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On perturbations of singular complex analytic curves Nandi, Achinta Kumar Complex Variables 32H35 (Primary), 32S05, 32B10 (Secondary) 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. |
| title | On perturbations of singular complex analytic curves |
| topic | Complex Variables 32H35 (Primary), 32S05, 32B10 (Secondary) |
| url | https://arxiv.org/abs/2307.11656 |