Holomorphic Mappings between Hyperquadrics with Small Signature Difference

TL;DR

This paper investigates holomorphic mappings between hyperquadrics with small signature differences, showing they can be normalized to simple forms and their images are contained in low-dimensional complex planes, with applications of a Hopf lemma type result.

Contribution

It establishes normalization and image dimension bounds for holomorphic hyperquadric mappings with small signature differences, extending understanding of their structure.

Findings

Mappings can be normalized to simple forms

Images are contained in low-dimensional complex planes

A Hopf lemma type result is proved

Abstract

In this paper, we study holomorphic mappings sending a hyperquadric of signature $\ell$ in $\bC^n$ into a hyperquadric of signature $\ell'$ in $\bC^N$. We show (Theorem \ref{main}) that if the signature difference $\ell'-\ell$ is not too large, then the mapping can be normalized by automorphisms of the target hyperquadric to a particularly simple form and, in particular, the image of the mapping is contained in a complex plane of a dimension that depends only on $\ell$ and $\ell'$, and not on the target dimension $N$. We also prove a Hopf Lemma type result (Theorem \ref{main2}) for such mappings.