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Article: Nonexistence of holomorphic submersions between complex unit balls equivariant with respect to a lattice and their generalizations

TitleNonexistence of holomorphic submersions between complex unit balls equivariant with respect to a lattice and their generalizations
Authors
Issue Date2010
PublisherThe Johns Hopkins University Press. The Journal's web site is located at http://www.press.jhu.edu/journals/american_journal_of_mathematics/index.html
Citation
American Journal Of Mathematics, 2010, v. 132 n. 5, p. 1347-1363 How to Cite?
AbstractIn this article we prove first of all the nonexistence of holomorphic submersions other than covering maps between compact quotients of complex unit balls, with a proof that works equally well in a more general equivariant setting. For a non-equidimensional surjective holomorphic map between compact ball quotients, our method applies to show that the set of critical values must be nonempty and of codimension 1. In the equivariant setting the line of arguments extends to holomorphic mappings of maximal rank into the complex projective space or the complex Euclidean space, yielding in the latter case a lower bound on the dimension of the singular loci of certain holomorphic maps defined by integrating holomorphic 1-forms. In another direction, we extend the nonexistence statement on holomorphic submersions to the case of ball quotients of finite volume, provided that the target complex unit ball is of dimension m > 2, giving in particular a new proof that a local biholomorphism between noncompact m-ball quotients of finite volume must be a covering map whenever m > 2. Finally, combining our results with Hermitian metric rigidity, we show that any holomorphic submersion from a bounded symmetric domain into a complex unit ball equivariant with respect to a lattice must factor through a canonical projection to yield an automorphism of the complex unit ball, provided that either the lattice is cocompact or the ball is of dimension at least 2. © 2010 by The Johns Hopkins University Press.
Persistent Identifierhttp://hdl.handle.net/10722/135147
ISSN
2015 Impact Factor: 1.118
2015 SCImago Journal Rankings: 3.513
References

 

DC FieldValueLanguage
dc.contributor.authorKoziarz, Ven_HK
dc.contributor.authorMok, Nen_HK
dc.date.accessioned2011-07-27T01:29:06Z-
dc.date.available2011-07-27T01:29:06Z-
dc.date.issued2010en_HK
dc.identifier.citationAmerican Journal Of Mathematics, 2010, v. 132 n. 5, p. 1347-1363en_HK
dc.identifier.issn0002-9327en_HK
dc.identifier.urihttp://hdl.handle.net/10722/135147-
dc.description.abstractIn this article we prove first of all the nonexistence of holomorphic submersions other than covering maps between compact quotients of complex unit balls, with a proof that works equally well in a more general equivariant setting. For a non-equidimensional surjective holomorphic map between compact ball quotients, our method applies to show that the set of critical values must be nonempty and of codimension 1. In the equivariant setting the line of arguments extends to holomorphic mappings of maximal rank into the complex projective space or the complex Euclidean space, yielding in the latter case a lower bound on the dimension of the singular loci of certain holomorphic maps defined by integrating holomorphic 1-forms. In another direction, we extend the nonexistence statement on holomorphic submersions to the case of ball quotients of finite volume, provided that the target complex unit ball is of dimension m > 2, giving in particular a new proof that a local biholomorphism between noncompact m-ball quotients of finite volume must be a covering map whenever m > 2. Finally, combining our results with Hermitian metric rigidity, we show that any holomorphic submersion from a bounded symmetric domain into a complex unit ball equivariant with respect to a lattice must factor through a canonical projection to yield an automorphism of the complex unit ball, provided that either the lattice is cocompact or the ball is of dimension at least 2. © 2010 by The Johns Hopkins University Press.en_HK
dc.languageengen_US
dc.publisherThe Johns Hopkins University Press. The Journal's web site is located at http://www.press.jhu.edu/journals/american_journal_of_mathematics/index.htmlen_HK
dc.relation.ispartofAmerican Journal of Mathematicsen_HK
dc.titleNonexistence of holomorphic submersions between complex unit balls equivariant with respect to a lattice and their generalizationsen_HK
dc.typeArticleen_HK
dc.identifier.emailMok, N:nmok@hkucc.hku.hken_HK
dc.identifier.authorityMok, N=rp00763en_HK
dc.description.naturelink_to_subscribed_fulltext-
dc.identifier.doi10.1353/ajm.2010.0007en_HK
dc.identifier.scopuseid_2-s2.0-77958584042en_HK
dc.identifier.hkuros186040en_US
dc.relation.referenceshttp://www.scopus.com/mlt/select.url?eid=2-s2.0-77958584042&selection=ref&src=s&origin=recordpageen_HK
dc.identifier.volume132en_HK
dc.identifier.issue5en_HK
dc.identifier.spage1347en_HK
dc.identifier.epage1363en_HK
dc.identifier.eissn1080-6377-
dc.publisher.placeUnited Statesen_HK
dc.identifier.scopusauthoridKoziarz, V=8352624600en_HK
dc.identifier.scopusauthoridMok, N=7004348032en_HK

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