On the CR transversality of holomorphic maps into hyperquadrics

TL;DR

This paper proves that holomorphic maps from certain Levi-nondegenerate hypersurfaces into hyperquadrics are necessarily CR transversal unless they are trivial, establishing a key embedding property in complex geometry.

Contribution

It demonstrates that under specific conditions, holomorphic maps into hyperquadrics must be CR transversal, extending understanding of CR embeddings and transversality in complex analysis.

Findings

Holomorphic maps are CR transversal unless they map a neighborhood into the hyperquadric.

Such maps are necessarily local CR embeddings.

The result applies to hypersurfaces with signature  in complex spaces.

Abstract

Let $M_\ell$ be a smooth Levi-nondegenerate hypersurface of signature $\ell$ in $\mathbf C^n$ with $ n\ge 3$, and write $H_\ell^N$ for the standard hyperquadric of the same signature in $\mathbf C^N$ with $N-n< \frac{n-1}{2}$. Let $F$ be a holomorphic map sending $M_\ell$ into $H_\ell^N$. Assume $F$ does not send a neighborhood of $M_\ell$ in $\mathbf C^n$ into $H_\ell^N$. We show that $F$ is necessarily CR transversal to $M_\ell$ at any point. Equivalently, we show that $F$ is a local CR embedding from $M_\ell$ into $H_\ell^N$.