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Article: Nonexistence of proper holomorphic maps between certain classical bounded symmetric domains

TitleNonexistence of proper holomorphic maps between certain classical bounded symmetric domains
Authors
KeywordsBounded symmetric domains
Characteristic symmetric subspaces
Invariantly geodesic subspaces
Proper holomorphic maps
Rank defects
Issue Date2008
PublisherWorld Scientific Publishing Co Pte Ltd. The Journal's web site is located at http://www.worldscinet.com/cam/cam.shtml
Citation
Chinese Annals Of Mathematics. Series B, 2008, v. 29 n. 2, p. 135-146 How to Cite?
AbstractThe author, motivated by his results on Hermitian metric rigidity, conjectured in [4] 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. The Conjecture was resolved in the affirmative by I.-H. Tsai [8]. When the hypothesis r′ ≤ r is removed, the structure of proper holomorphic maps f : Ω → Ω′ is far from being understood, and the complexity in studying such maps depends very much on the difference r′ - r, which is called the rank defect. The only known nontrivial non-equidimensional structure theorems on proper holomorphic maps are due to Z.-H. Tu [10], in which a rigidity theorem was proven for certain pairs of classical domains of type I, which implies nonexistence theorems for other pairs of such domains. For both results the rank defect is equal to 1, and a generalization of the rigidity result to cases of higher rank defects along the line of arguments of [10] has so far been inaccessible. In this article, the author produces nonexistence results for infinite series of pairs of (Ω, Ω′) of irreducible bounded symmetric domains of type I in which the rank defect is an arbitrarily prescribed positive integer. Such nonexistence results are obtained by exploiting the geometry of characteristic symmetric subspaces as introduced by N. Mok and I.-H Tsai [6] and more generally invariantly geodesic subspaces as formalized in [8]. Our nonexistence results motivate the formulation of questions on proper holomorphic maps in the non-equirank case. © 2008 Editorial Office of CAM (Fudan University) and Springer-Verlag Berlin Heidelberg.
Persistent Identifierhttp://hdl.handle.net/10722/75390
ISSN
2015 Impact Factor: 0.452
2015 SCImago Journal Rankings: 0.445
ISI Accession Number ID
References

 

DC FieldValueLanguage
dc.contributor.authorMok, Nen_HK
dc.date.accessioned2010-09-06T07:10:40Z-
dc.date.available2010-09-06T07:10:40Z-
dc.date.issued2008en_HK
dc.identifier.citationChinese Annals Of Mathematics. Series B, 2008, v. 29 n. 2, p. 135-146en_HK
dc.identifier.issn0252-9599en_HK
dc.identifier.urihttp://hdl.handle.net/10722/75390-
dc.description.abstractThe author, motivated by his results on Hermitian metric rigidity, conjectured in [4] 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. The Conjecture was resolved in the affirmative by I.-H. Tsai [8]. When the hypothesis r′ ≤ r is removed, the structure of proper holomorphic maps f : Ω → Ω′ is far from being understood, and the complexity in studying such maps depends very much on the difference r′ - r, which is called the rank defect. The only known nontrivial non-equidimensional structure theorems on proper holomorphic maps are due to Z.-H. Tu [10], in which a rigidity theorem was proven for certain pairs of classical domains of type I, which implies nonexistence theorems for other pairs of such domains. For both results the rank defect is equal to 1, and a generalization of the rigidity result to cases of higher rank defects along the line of arguments of [10] has so far been inaccessible. In this article, the author produces nonexistence results for infinite series of pairs of (Ω, Ω′) of irreducible bounded symmetric domains of type I in which the rank defect is an arbitrarily prescribed positive integer. Such nonexistence results are obtained by exploiting the geometry of characteristic symmetric subspaces as introduced by N. Mok and I.-H Tsai [6] and more generally invariantly geodesic subspaces as formalized in [8]. Our nonexistence results motivate the formulation of questions on proper holomorphic maps in the non-equirank case. © 2008 Editorial Office of CAM (Fudan University) and Springer-Verlag Berlin Heidelberg.en_HK
dc.languageengen_HK
dc.publisherWorld Scientific Publishing Co Pte Ltd. The Journal's web site is located at http://www.worldscinet.com/cam/cam.shtmlen_HK
dc.relation.ispartofChinese Annals of Mathematics. Series Ben_HK
dc.rightsCreative Commons: Attribution 3.0 Hong Kong License-
dc.subjectBounded symmetric domainsen_HK
dc.subjectCharacteristic symmetric subspacesen_HK
dc.subjectInvariantly geodesic subspacesen_HK
dc.subjectProper holomorphic mapsen_HK
dc.subjectRank defectsen_HK
dc.titleNonexistence of proper holomorphic maps between certain classical bounded symmetric domainsen_HK
dc.typeArticleen_HK
dc.identifier.emailMok, N:nmok@hkucc.hku.hken_HK
dc.identifier.authorityMok, N=rp00763en_HK
dc.description.naturepostprint-
dc.identifier.doi10.1007/s11401-007-0174-3en_HK
dc.identifier.scopuseid_2-s2.0-43949125519en_HK
dc.identifier.hkuros142688en_HK
dc.relation.referenceshttp://www.scopus.com/mlt/select.url?eid=2-s2.0-43949125519&selection=ref&src=s&origin=recordpageen_HK
dc.identifier.volume29en_HK
dc.identifier.issue2en_HK
dc.identifier.spage135en_HK
dc.identifier.epage146en_HK
dc.identifier.isiWOS:000254021900003-
dc.publisher.placeSingaporeen_HK
dc.identifier.scopusauthoridMok, N=7004348032en_HK

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