On holomorphic curves tangent to real hypersurfaces of infinite type

TL;DR

This paper explores the geometric structure of infinite type real hypersurfaces in complex space, providing conditions for the existence of holomorphic curves tangent to these hypersurfaces to infinite order using Newton polyhedra.

Contribution

It introduces new criteria involving Newton polyhedra for the existence of tangent holomorphic curves to infinite type hypersurfaces, especially for certain model cases.

Findings

Established a sufficient condition for tangent holomorphic curves using Newton polyhedra.

Provided equivalence conditions for some model hypersurfaces.

Analyzed the flatness and geometric properties of infinite type hypersurfaces.

Abstract

The purpose of this paper is to investigate the geometric properties of real hypersurfaces of D'Angelo infinite type in ${\mathbb C}^n$. In order to understand the situation of flatness of these hypersurfaces, it is natural to ask whether there exists a nonconstant holomorphic curve tangent to a given hypersurface to infinite order. A sufficient condition for this existence is given by using Newton polyhedra,which is an important concept in singularity theory. More precisely,equivalence conditions are given in the case of some model hypersurfaces.