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Holomorphic mappings between hyperquadrics with small signature difference
- American Journal of Mathematics
- Johns Hopkins University Press
- Volume 133, Number 6, December 2011
- pp. 1633-1661
- 10.1353/ajm.2011.0044
- Article
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In this paper, we study holomorphic mappings sending a hyperquadric of
signature $\ell$ in ${\Bbb C}^n$ into a hyperquadric of signature $\ell'$
in ${\Bbb C}^N$. We show (Theorem 1.1) 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
1.3) for such mappings.