Complex Geometry and Dynamics pp 139-155 | Cite as

# On the CR Transversality of Holomorphic Maps into Hyperquadrics

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## Abstract

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

## Keywords

Power Series Expansion Open Dense Subset Quantitative Version Weighted Degree Rigidity Problem## Notes

### Acknowledgements

Part of work was done when the authors were attending the 7th Workshop on geometric analysis of PDE and several complex variables at Serra Negra, Brazil. Both authors would like to thank the organizers for the kind invitation. The second author is also indebted to Wuhan University for the hospitality during her several visits there.

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