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On perturbations of singular complex analytic curves

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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 and Peters (Geom. Funct. Anal. 24, 887–915 (2014)).

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Acknowledgements

The author would like to acknowledge Jiří Lebl for introducing the result by Lyubich-Peters, and for his kind and patient guidance throughout. The author is grateful to Roland Roeder for carefully reading the manuscript and suggesting necessary corrections and for many long mathematical discussions. His inputs have greatly helped in finding cleaner proofs of many results in this article. The author is thankful to Anand Patel, Sean Curry, Abdullah Al Helal, and Anthony Kable for helpful discussions on various results of this article.

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  1. Department of Mathematics, Oklahoma State University, Stillwater, OK, 74078, USA

    Achinta Kumar Nandi

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  1. Achinta Kumar Nandi
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The author proved all the theorems and constructed the results on his own. The author wrote the manuscript and received feedback from his mentors.

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Correspondence to Achinta Kumar Nandi.

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Nandi, A.K. On perturbations of singular complex analytic curves. Complex Anal Synerg 10, 7 (2024). https://doi.org/10.1007/s40627-024-00133-1

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