Local and infinitesimal rigidity of hypersurface embeddings

TL;DR

This paper investigates the local rigidity of holomorphic hypersurface embeddings in complex spaces, demonstrating that infinitesimal conditions can ensure actual rigidity, especially for generic embeddings into hyperquadrics.

Contribution

It establishes that infinitesimal conditions lead to local rigidity in hypersurface embeddings and proves generic embeddings into hyperquadrics are locally rigid.

Findings

Infinitesimal conditions imply local rigidity in certain cases

Generic embeddings into hyperquadrics are locally rigid

Holomorphic hypersurface embeddings exhibit specific rigidity properties

Abstract

We study local rigidity properties of holomorphic embeddings of real hypersurfaces in $\mathbb C^2$ into real hypersurfaces in $\mathbb C^3$ and show that infinitesimal conditions imply actual local rigidity in a number of (important) cases. We use this to show that generic embeddings into a hyperquadric in $\mathbb C^3$ are locally rigid.