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Article: Holomorphic retractions of bounded symmetric domains onto totally geodesic complex submanifolds

TitleHolomorphic retractions of bounded symmetric domains onto totally geodesic complex submanifolds
Authors
Keywords17B22
32H02
53C35
Bounded symmetric domain
Harish-Chandra embedding
Holomorphic retraction
Totally geodesy
Issue Date31-Mar-2023
PublisherSpringer
Citation
Chinese Annals of Mathematics, Series B, 2023, v. 43, n. 6, p. 1125-1142 How to Cite?
Abstract

Given a bounded symmetric domain Ω the author considers the geometry of its totally geodesic complex submanifolds S ⊂ Ω. In terms of the Harish-Chandra realization Ω ⋐ ℂn and taking S to pass through the origin 0 ∈ Ω, so that S = E ⋂ Ω for some complex vector subspace of ℂn, the author shows that the orthogonal projection ρ: Ω → E maps Ω onto S, and deduces that S ⊂ Ω is a holomorphic isometry with respect to the Carathéodory metric. His first theorem gives a new derivation of a result of Yeung’s deduced from the classification theory by Satake and Ihara in the special case of totally geodesic complex submanifolds of rank 1 and of complex dimension ≥ 2 in the Siegel upper half plane ��, a result which was crucial for proving the nonexistence of totally geodesic complex suborbifolds of dimension ≥ 2 on the open Torelli locus of the Siegel modular variety �� by the same author. The proof relies on the characterization of totally geodesic submanifolds of Riemannian symmetric spaces in terms of Lie triple systems and a variant of the Hermann Convexity Theorem giving a new characterization of the Harish-Chandra realization in terms of bisectional curvatures.


Persistent Identifierhttp://hdl.handle.net/10722/339305
ISSN
2023 Impact Factor: 0.5
2023 SCImago Journal Rankings: 0.237
ISI Accession Number ID

 

DC FieldValueLanguage
dc.contributor.authorMok, Ngaiming-
dc.date.accessioned2024-03-11T10:35:33Z-
dc.date.available2024-03-11T10:35:33Z-
dc.date.issued2023-03-31-
dc.identifier.citationChinese Annals of Mathematics, Series B, 2023, v. 43, n. 6, p. 1125-1142-
dc.identifier.issn0252-9599-
dc.identifier.urihttp://hdl.handle.net/10722/339305-
dc.description.abstract<p>Given a bounded symmetric domain Ω the author considers the geometry of its totally geodesic complex submanifolds <em>S</em> ⊂ Ω. In terms of the Harish-Chandra realization Ω ⋐ ℂ<sup><em>n</em></sup> and taking <em>S</em> to pass through the origin 0 ∈ Ω, so that <em>S</em> = <em>E</em> ⋂ Ω for some complex vector subspace of ℂ<sup><em>n</em></sup>, the author shows that the orthogonal projection <em>ρ</em>: Ω → <em>E</em> maps Ω onto <em>S</em>, and deduces that <em>S</em> ⊂ Ω is a holomorphic isometry with respect to the Carathéodory metric. His first theorem gives a new derivation of a result of Yeung’s deduced from the classification theory by Satake and Ihara in the special case of totally geodesic complex submanifolds of rank 1 and of complex dimension ≥ 2 in the Siegel upper half plane ��, a result which was crucial for proving the nonexistence of totally geodesic complex suborbifolds of dimension ≥ 2 on the open Torelli locus of the Siegel modular variety �� by the same author. The proof relies on the characterization of totally geodesic submanifolds of Riemannian symmetric spaces in terms of Lie triple systems and a variant of the Hermann Convexity Theorem giving a new characterization of the Harish-Chandra realization in terms of bisectional curvatures.<br></p>-
dc.languageeng-
dc.publisherSpringer-
dc.relation.ispartofChinese Annals of Mathematics, Series B-
dc.subject17B22-
dc.subject32H02-
dc.subject53C35-
dc.subjectBounded symmetric domain-
dc.subjectHarish-Chandra embedding-
dc.subjectHolomorphic retraction-
dc.subjectTotally geodesy-
dc.titleHolomorphic retractions of bounded symmetric domains onto totally geodesic complex submanifolds-
dc.typeArticle-
dc.identifier.doi10.1007/s11401-022-0380-z-
dc.identifier.scopuseid_2-s2.0-85143513705-
dc.identifier.volume43-
dc.identifier.issue6-
dc.identifier.spage1125-
dc.identifier.epage1142-
dc.identifier.eissn1860-6261-
dc.identifier.isiWOS:000895451200010-
dc.identifier.issnl0252-9599-

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