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Article: Characterization of standard embeddings between complex Grassmannians by means of varieties of minimal rational tangents

TitleCharacterization of standard embeddings between complex Grassmannians by means of varieties of minimal rational tangents
Authors
KeywordsAnalytic Continuation
Bounded Symmetric Domains
Minimal Rational Curves
Proper Holomorphic Maps
Rigidity
Varieties Of Minimal Rational Tangents
Issue Date2008
PublisherScience in China Press. The Journal's web site is located at http://www.scichina.com:8081/sciAe/EN/volumn/current.shtml
Citation
Science In China, Series A: Mathematics, 2008, v. 51 n. 4, p. 660-684 How to Cite?
AbstractIn 1993, Tsai proved that a proper holomorphic mapping f: Ω → Ω′ from an irreducible bounded symmetric domain Ω of rank ≥ 2 into a bounded symmetric domain Ω′is necessarily totally geodesic provided that r′:= rank(Ω′) ≤ rank(Ω):= r, proving a conjecture of the author's motivated by Hermitian metric rigidity. As a first step in the proof, Tsai showed that df preserves almost everywhere the set of tangent vectors of rank 1. Identifying bounded symmetric domains as open subsets of their compact duals by means of the Borel embedding, this means that the germ of f at a general point preserves the varieties of minimal rational tangents (VMRTs). In another completely different direction Hwang-Mok established with very few exceptions the Cartan-Fubini extension priniciple for germs of local biholomorphisms between Fano manifolds of Picard number 1, showing that the germ of map extends to a global biholomorphism provided that it preserves VMRTs. We propose to isolate the problem of characterization of special holomorphic embeddings between Fano manifolds of Picard number 1, especially in the case of classical manifolds such as rational homogeneous spaces of Picard number 1, by a non-equidimensional analogue of the Cartan-Fubini extension principle. As an illustration we show along this line that standard embeddings between complex Grassmann manifolds of rank ≤ 2 can be characterized by the VMRT-preserving property and a non-degeneracy condition, giving a new proof of a result of Neretin's which on the one hand paves the way for far-reaching generalizations to the context of rational homogeneous spaces and more generally Fano manifolds of Picard number 1, on the other hand should be applicable to the study of proper holomorphic mappings between bounded domains carrying some form of geometric structures. © 2008 Science Press.
Persistent Identifierhttp://hdl.handle.net/10722/156209
ISSN
2011 Impact Factor: 0.701
2012 SCImago Journal Rankings: 0.540
ISI Accession Number ID
References

 

DC FieldValueLanguage
dc.contributor.authorMok, Nen_US
dc.date.accessioned2012-08-08T08:40:51Z-
dc.date.available2012-08-08T08:40:51Z-
dc.date.issued2008en_US
dc.identifier.citationScience In China, Series A: Mathematics, 2008, v. 51 n. 4, p. 660-684en_US
dc.identifier.issn1006-9283en_US
dc.identifier.urihttp://hdl.handle.net/10722/156209-
dc.description.abstractIn 1993, Tsai proved that a proper holomorphic mapping f: Ω → Ω′ from an irreducible bounded symmetric domain Ω of rank ≥ 2 into a bounded symmetric domain Ω′is necessarily totally geodesic provided that r′:= rank(Ω′) ≤ rank(Ω):= r, proving a conjecture of the author's motivated by Hermitian metric rigidity. As a first step in the proof, Tsai showed that df preserves almost everywhere the set of tangent vectors of rank 1. Identifying bounded symmetric domains as open subsets of their compact duals by means of the Borel embedding, this means that the germ of f at a general point preserves the varieties of minimal rational tangents (VMRTs). In another completely different direction Hwang-Mok established with very few exceptions the Cartan-Fubini extension priniciple for germs of local biholomorphisms between Fano manifolds of Picard number 1, showing that the germ of map extends to a global biholomorphism provided that it preserves VMRTs. We propose to isolate the problem of characterization of special holomorphic embeddings between Fano manifolds of Picard number 1, especially in the case of classical manifolds such as rational homogeneous spaces of Picard number 1, by a non-equidimensional analogue of the Cartan-Fubini extension principle. As an illustration we show along this line that standard embeddings between complex Grassmann manifolds of rank ≤ 2 can be characterized by the VMRT-preserving property and a non-degeneracy condition, giving a new proof of a result of Neretin's which on the one hand paves the way for far-reaching generalizations to the context of rational homogeneous spaces and more generally Fano manifolds of Picard number 1, on the other hand should be applicable to the study of proper holomorphic mappings between bounded domains carrying some form of geometric structures. © 2008 Science Press.en_US
dc.languageengen_US
dc.publisherScience in China Press. The Journal's web site is located at http://www.scichina.com:8081/sciAe/EN/volumn/current.shtmlen_US
dc.relation.ispartofScience in China, Series A: Mathematicsen_US
dc.subjectAnalytic Continuationen_US
dc.subjectBounded Symmetric Domainsen_US
dc.subjectMinimal Rational Curvesen_US
dc.subjectProper Holomorphic Mapsen_US
dc.subjectRigidityen_US
dc.subjectVarieties Of Minimal Rational Tangentsen_US
dc.titleCharacterization of standard embeddings between complex Grassmannians by means of varieties of minimal rational tangentsen_US
dc.typeArticleen_US
dc.identifier.emailMok, N:nmok@hkucc.hku.hken_US
dc.identifier.authorityMok, N=rp00763en_US
dc.description.naturelink_to_subscribed_fulltexten_US
dc.identifier.doi10.1007/s11425-008-0005-3en_US
dc.identifier.scopuseid_2-s2.0-42149126672en_US
dc.identifier.hkuros141187-
dc.relation.referenceshttp://www.scopus.com/mlt/select.url?eid=2-s2.0-42149126672&selection=ref&src=s&origin=recordpageen_US
dc.identifier.volume51en_US
dc.identifier.issue4en_US
dc.identifier.spage660en_US
dc.identifier.epage684en_US
dc.identifier.isiWOS:000254626300013-
dc.publisher.placeChinaen_US
dc.identifier.scopusauthoridMok, N=7004348032en_US

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