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Article: Factorization of proper holomorphic maps on irreducible bounded symmetric domains of rank ≥ 2
Title | Factorization of proper holomorphic maps on irreducible bounded symmetric domains of rank ≥ 2 | ||||||
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Authors | |||||||
Keywords | Bounded symmetric domain Correspondence Discriminant Fatou's theorem G-structure Proper holomorphic map | ||||||
Issue Date | 2010 | ||||||
Publisher | Science China Press, co-published with Springer. The Journal's web site is located at http://math.scichina.com/english/ | ||||||
Citation | Science China Mathematics, 2010, v. 53 n. 3, p. 813-826 How to Cite? | ||||||
Abstract | We obtain rigidity results on arbitrary proper holomorphic maps F from an irreducible bounded symmetric domain Ω of rank ≥ 2 into any complex space Z. After lifting to the normalization of the subvariety F(Ω) ⊂ Z, we prove that F must be the canonical projection map to the quotient space of Ω by a finite group of automorphisms. The approach is along the line of the works of Mok and Tsai by considering radial limits of bounded holomorphic functions derived from F and proving that proper holomorphic maps between bounded symmetric domains preserve certain totally geodesic subdomains. In contrast to the previous works, in general we have to deal with multivalent holomorphic maps for which Fatou's theorem cannot be applied directly. We bypass the difficulty by devising a limiting process for taking radial limits of correspondences arising from proper holomorphic maps and by elementary estimates allowing us to define distinct univalent branches of the underlying multivalent map on certain subsets. As a consequence of our rigidity result, with the exception of Type-IV domains, any proper holomorphic map f: Ω → D of Ω onto a bounded convex domain D is necessarily a biholomorphism. In the exceptional case where Ω is a Type-IV domain, either f is a biholomorphism or it is a double cover branched over a totally geodesic submanifold which can be explicitly described. © Science China Press and Springer-Verlag Berlin Heidelberg 2010. | ||||||
Persistent Identifier | http://hdl.handle.net/10722/75302 | ||||||
ISSN | 2023 Impact Factor: 1.4 2023 SCImago Journal Rankings: 1.060 | ||||||
ISI Accession Number ID |
Funding Information: This work was partially supported by the GRF 7032/08P of the HKRGC, Hong Kong and National Natural Science Foundation of China (Grant No. 10971156). The first author wishes to thank the organizers of the International Conference on Complex Analysis and Related Topics, held August 2009 at the Chinese Academy of Sciences, Beijing, especially Professor Wang Yuefei, for their kind invitation and for the opportunity to take part in the memorable event in honor of Professor Yang Lo. The authors wish to dedicate this article to Professor Yang Lo on the occasion of his 70th birthday. | ||||||
References |
DC Field | Value | Language |
---|---|---|
dc.contributor.author | Mok, N | en_HK |
dc.contributor.author | Ng, SC | en_HK |
dc.contributor.author | Tu, ZH | en_HK |
dc.date.accessioned | 2010-09-06T07:09:50Z | - |
dc.date.available | 2010-09-06T07:09:50Z | - |
dc.date.issued | 2010 | en_HK |
dc.identifier.citation | Science China Mathematics, 2010, v. 53 n. 3, p. 813-826 | en_HK |
dc.identifier.issn | 1674-7283 | en_HK |
dc.identifier.uri | http://hdl.handle.net/10722/75302 | - |
dc.description.abstract | We obtain rigidity results on arbitrary proper holomorphic maps F from an irreducible bounded symmetric domain Ω of rank ≥ 2 into any complex space Z. After lifting to the normalization of the subvariety F(Ω) ⊂ Z, we prove that F must be the canonical projection map to the quotient space of Ω by a finite group of automorphisms. The approach is along the line of the works of Mok and Tsai by considering radial limits of bounded holomorphic functions derived from F and proving that proper holomorphic maps between bounded symmetric domains preserve certain totally geodesic subdomains. In contrast to the previous works, in general we have to deal with multivalent holomorphic maps for which Fatou's theorem cannot be applied directly. We bypass the difficulty by devising a limiting process for taking radial limits of correspondences arising from proper holomorphic maps and by elementary estimates allowing us to define distinct univalent branches of the underlying multivalent map on certain subsets. As a consequence of our rigidity result, with the exception of Type-IV domains, any proper holomorphic map f: Ω → D of Ω onto a bounded convex domain D is necessarily a biholomorphism. In the exceptional case where Ω is a Type-IV domain, either f is a biholomorphism or it is a double cover branched over a totally geodesic submanifold which can be explicitly described. © Science China Press and Springer-Verlag Berlin Heidelberg 2010. | en_HK |
dc.language | eng | en_HK |
dc.publisher | Science China Press, co-published with Springer. The Journal's web site is located at http://math.scichina.com/english/ | en_HK |
dc.relation.ispartof | Science China Mathematics | en_HK |
dc.subject | Bounded symmetric domain | en_HK |
dc.subject | Correspondence | en_HK |
dc.subject | Discriminant | en_HK |
dc.subject | Fatou's theorem | en_HK |
dc.subject | G-structure | en_HK |
dc.subject | Proper holomorphic map | en_HK |
dc.title | Factorization of proper holomorphic maps on irreducible bounded symmetric domains of rank ≥ 2 | en_HK |
dc.type | Article | en_HK |
dc.identifier.email | Mok, N:nmok@hkucc.hku.hk | en_HK |
dc.identifier.authority | Mok, N=rp00763 | en_HK |
dc.description.nature | link_to_OA_fulltext | - |
dc.identifier.doi | 10.1007/s11425-010-0058-y | en_HK |
dc.identifier.scopus | eid_2-s2.0-77952175839 | en_HK |
dc.identifier.hkuros | 169796 | en_HK |
dc.relation.references | http://www.scopus.com/mlt/select.url?eid=2-s2.0-77952175839&selection=ref&src=s&origin=recordpage | en_HK |
dc.identifier.volume | 53 | en_HK |
dc.identifier.issue | 3 | en_HK |
dc.identifier.spage | 813 | en_HK |
dc.identifier.epage | 826 | en_HK |
dc.identifier.isi | WOS:000276597700028 | - |
dc.publisher.place | China | en_HK |
dc.identifier.scopusauthorid | Mok, N=7004348032 | en_HK |
dc.identifier.scopusauthorid | Ng, SC=35264831700 | en_HK |
dc.identifier.scopusauthorid | Tu, ZH=13702842300 | en_HK |
dc.identifier.issnl | 1869-1862 | - |