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A note on complex plane curve singularities up to diffeomorphism and their rigidity

A. Fernández-Hernández, R. Giménez Conejero

Abstract

We prove that, if two germs of plane curves $(C,0)$ and $(C',0)$ with at least one singular branch are equivalent by a (real) smooth diffeomorphism, then $C$ is complex isomorphic to $C'$ or to $\overline{C'}$. A similar result was shown by Ephraim for irreducible hypersurfaces before, but his proof is not constructive. Indeed, we show that the complex isomorphism is given by the Taylor series of the diffeomorphism. We also prove an analogous result for the case of non-irreducible hypersurfaces containing an irreducible component of zero-dimensional isosingular locus. Moreover, we provide a general overview of the different classifications of plane curve singularities.

A note on complex plane curve singularities up to diffeomorphism and their rigidity

Abstract

We prove that, if two germs of plane curves and with at least one singular branch are equivalent by a (real) smooth diffeomorphism, then is complex isomorphic to or to . A similar result was shown by Ephraim for irreducible hypersurfaces before, but his proof is not constructive. Indeed, we show that the complex isomorphism is given by the Taylor series of the diffeomorphism. We also prove an analogous result for the case of non-irreducible hypersurfaces containing an irreducible component of zero-dimensional isosingular locus. Moreover, we provide a general overview of the different classifications of plane curve singularities.
Paper Structure (17 sections, 21 theorems, 64 equations, 3 figures)

This paper contains 17 sections, 21 theorems, 64 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Different classifications
  3. Topological
  4. Bilipschitz
  5. Real analytic and smooth
  6. Complex analytic
  7. Other classifications
  8. Preliminaries on formal power series
  9. Formal power series
  10. Formal parametrizations
  11. Main result
  12. Simplification to linear terms
  13. Holomorphy
  14. Proof
  15. Curves with only smooth branches
  16. ...and 2 more sections

Key Result

Theorem 1

$(C,0)$ and $(C',0)$ are diffeomorphic if, and only if, they are complex isomorphic or $C$ is complex isomorphic to $\overline{C'}$, which is also a complex curve with equation $\overline{g'(\overline{x},\overline{y})}=0$. Furthermore, any diffeomorphism $\psi$ that takes one plane curve singularity

Figures (3)

  • Figure 1: General overview of different classifications of plane curve singularities.
  • Figure 2: Representation of two plane curve singularities, one irreducible (left) and the other with two branches (right). Observe the conical structure over their link.
  • Figure 3: The diffeomorphism $\psi$ of \ref{['eq:difeoexample']}.

Theorems & Definitions (49)

  • Theorem
  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3
  • Remark 2.4
  • Definition 2.5
  • Theorem 2.6
  • Proposition 2.7
  • Remark 2.8
  • Definition 2.9
  • ...and 39 more