Local equivalence of symmetric hypersurfaces in $\mathbb C^2$

TL;DR

This paper develops a complete normalization method for symmetric Levi degenerate hypersurfaces in ^2, respecting symmetries, and classifies biholomorphic maps between tubes, advancing the understanding of local equivalence in complex geometry.

Contribution

It introduces a symmetry-respecting normal form for symmetric Levi degenerate hypersurfaces in ^2, enabling classification and analysis of biholomorphic maps.

Findings

Complete normalization for symmetric Levi degenerate hypersurfaces.

Classification of biholomorphic maps between tubes.

Effective tools for symmetry-preserving equivalence analysis.

Abstract

The Chern-Moser normal form and its analog on finite type hypersurfaces in general do not respect symmetries. Extending the work of N. K. Stanton, we consider the local equivalence problem for symmetric Levi degenerate hypersurfaces of finite type in $\mathbb C^2$. The results give for all such hypersurfaces a complete normalization which respects the symmetries. In particular, they apply to tubes and rigid hypersurfaces, providing an effective classification. The main tool is a complete normal form constructed for a general hypersurface with a tube model. As an application, we describe all biholomorphic maps between tubes, answering a question posed by N. Hanges.