Super-rigidity for holomorphic Segre maps

Abstract

Let ${\mathcal{H}}_l^n$ and ${\mathcal{H}}_{l,l^{\prime },n}^N$ be the complexifications of the real hyperquadrics in ${\mathbb{C}}^n$ and ${\mathbb{C}}^N$ of signatures l and $l^{\prime }$, respectively. We study local holomorphic Segre maps sending a piece of ${\mathcal{H}}_l^n$ into ${\mathcal{H}}_{l,l^{\prime },n}^N$ with the degeneracy rank $l_0=1$ and $N\ge n$. We show that these maps possess super rigidity properties.

Keywords:

• Holomorphic Segre maps
• hyperquadrics
• super rigidity
• degeneracy rank

2020 Mathematics Subject Classifications:

• 32H02
• 32H35

Acknowledgements

The authors would like to thank the referees for many valuable suggestions which have greatly improved both the exposition and the mathematics of the paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

All the authors are supported is part by National Natural Science Foundation of China (grant number 12171372).