A normal form for 1-infinite type hypersurfaces in $\mathbb C^2$. I. Formal Theory

TL;DR

This paper develops a formal normal form for infinite type hypersurfaces in complex two-space, revealing resonances and automorphism properties, and closely parallels the classical Chern-Moser form at Levi-nondegenerate points.

Contribution

It introduces a complete formal normal form for infinite type hypersurfaces with minimal degeneracy, highlighting the role of resonances and automorphism determination by jets.

Findings

Normal form resembles Chern-Moser form at Levi-nondegenerate points

Automorphisms are determined by 1-jets in the absence of resonances

Examples show high resonances lead to automorphisms with identical jets up to resonant order

Abstract

In this paper, we study the real hypersurfaces $M$ in $\mathbb C^2$ at points $p\in M$ of infinite type. The degeneracy of $M$ at $p$ is assumed to be the least possible, namely such that the Levi form vanishes to first order in the CR transversal direction. A new phenomenon, compared to known normal forms in other cases, is the presence of resonances as roots of an universal polynomial in the $7$-jet of the defining function of $M$. The main result is a complete (formal) normal form at points $p$ with no resonances. Remarkably, our normal form at such infinite type points resembles closely the Chern-Moser normal form at Levi-nondegenerate points. For a fixed hypersurface, its normal forms are parametrized by $S^1\times \mathbb R^*$, and as a corollary we find that the automorphisms in the stability group of $M$ at $p$ without resonances are determined by their $1$-jets at $p$. In the last section, as a contrast, we also give examples of hypersurfaces with arbitrarily high resonances that possess families of distinct automorphisms whose jets agree up to the resonant order.