Holomorphic Mappings between Hyperquadrics with Small Signature Difference

Details Holomorphic Mappings between Hyperquadrics with Small Signature Difference Holomorphic Mappings between Hyperquadrics with Small Signature Difference

2009-06-06

arxiv.org/abs/0906.1235v1

Mathematics

Complex Variables

Scientific paper

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.

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