Convergent normal form and canonical connection for hypersurfaces of finite type in $\mathbb C^2$

TL;DR

This paper introduces a new convergent normal form and a canonical connection for finite type hypersurfaces in 2, providing a geometric condition for their existence and an explicit construction, including the first invariant connection for Levi-degenerate cases.

Contribution

It presents a geometric condition ensuring a convergent normal form and constructs a canonical connection for finite type hypersurfaces in 2, including Levi-degenerate cases.

Findings

Established a geometric condition for normal form existence

Constructed an explicit convergent normal form

Developed the first invariant connection for Levi-degenerate hypersurfaces

Abstract

We study the holomorphic equivalence problem for finite type hypersurfaces in $\mathbb C^2$. We discover a geometric condition, which is sufficient for the existence of a natural convergent normal form for a finite type hypersurface. We also provide an explicit construction of such a normal form. As an application, we construct a canonical connection for a large class of finite type hypersurfaces. To the best of our knowledge, this gives the first construction of an invariant connection for Levi-degenerate hypersurfaces in $\mathbb C^2$.