On transfer of biholomorphisms across nonminimal loci

TL;DR

This paper investigates conditions under which local biholomorphisms can be transferred across nonminimal loci in real analytic hypersurfaces in complex space, focusing on Levi form properties and model pseudospheres.

Contribution

It establishes criteria for transferring biholomorphic equivalences across nonminimal points in hypersurfaces with nondegenerate Levi form.

Findings

Biholomorphic transfer is possible under specific Levi form conditions.

Hypersurfaces are locally equivalent to Heisenberg pseudospheres with possibly different signatures.

The results connect local geometry with global biholomorphic properties.

Abstract

A connected real analytic hypersurface M in C^(n+1) whose Levi form is nondegenerate in at least one point - hence at every point of some Zariski-open subset - is locally biholomorphic to the model Heisenberg quadric pseudosphere of signature (k, n-k) in one point if and only if, at every other Levi nondegenerate point, it is also locally biholomorphic to some Heisenberg pseudosphere, possibly having different signature (l, n-l).