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Book Chapter: Projective Algebraicity of Minimal Compactifications of Complex-Hyperbolic Space Forms of Finite Volume

TitleProjective Algebraicity of Minimal Compactifications of Complex-Hyperbolic Space Forms of Finite Volume
Authors
Issue Date2012
PublisherBirkhäuser Springer
Citation
Projective Algebraicity of Minimal Compactifications of Complex-Hyperbolic Space Forms of Finite Volume. In Itenberg, I., Jöricke, B & Passare, M (Eds.), Perspectives in analysis, geometry, and topology: on the occasion of the 60th birthday of Oleg Viro, p. 331-354. New York: Birkhäuser Springer, 2012 How to Cite?
AbstractLet Ωbe a bounded symmetric domain and Γ⊂ Aut(Ω) be an irreducible nonuniform torsion-free discrete subgroup. When Γis of rank ≥ 2, Γis necessarily arithmetic, and X :=Ω/Γ?admits a Satake-Baily-Borel compactification. When Ω?is of rank 1, i.e., the complex unit ball Bn of dimension n≥ 1,Γmay be nonarithmetic.When n≥2, by a general result of Siu and Yau, X is pseudoconcave and it can be compactified to a Moishezon space by adding a finite number of normal isolated singularities. In this article we show that for X := Bn/Γ the latter compactification is in fact projective-algebraic.We do this by showing that, just as in the arithmetic case of rank-1, X admits a smooth toroidal compactification X¯ M obtained by adjoining an Abelian variety to each of its finitely many ends, and X¯ M can be blown down to a normal projective-algebraic variety X¯ Min by solving ∂¯ with L2-estimates with respect to the canonical Kähler-Einstein metric and by normalization. As an application, we give an alternative proof of results of Koziarz-Mok on the submersion problem in the case of complex-hyperbolic space forms of finite volume by adapting the cohomological arguments in the compact case to general hyperplane sections of the minimal projective-algebraic compactifications which avoid the isolated singularities.
Persistent Identifierhttp://hdl.handle.net/10722/187440
ISBN
ISI Accession Number ID
Series/Report no.Progress in mathematics; v.296

 

DC FieldValueLanguage
dc.contributor.authorMok, Nen_US
dc.date.accessioned2013-08-20T12:45:18Z-
dc.date.available2013-08-20T12:45:18Z-
dc.date.issued2012en_US
dc.identifier.citationProjective Algebraicity of Minimal Compactifications of Complex-Hyperbolic Space Forms of Finite Volume. In Itenberg, I., Jöricke, B & Passare, M (Eds.), Perspectives in analysis, geometry, and topology: on the occasion of the 60th birthday of Oleg Viro, p. 331-354. New York: Birkhäuser Springer, 2012en_US
dc.identifier.isbn9780817682767-
dc.identifier.urihttp://hdl.handle.net/10722/187440-
dc.description.abstractLet Ωbe a bounded symmetric domain and Γ⊂ Aut(Ω) be an irreducible nonuniform torsion-free discrete subgroup. When Γis of rank ≥ 2, Γis necessarily arithmetic, and X :=Ω/Γ?admits a Satake-Baily-Borel compactification. When Ω?is of rank 1, i.e., the complex unit ball Bn of dimension n≥ 1,Γmay be nonarithmetic.When n≥2, by a general result of Siu and Yau, X is pseudoconcave and it can be compactified to a Moishezon space by adding a finite number of normal isolated singularities. In this article we show that for X := Bn/Γ the latter compactification is in fact projective-algebraic.We do this by showing that, just as in the arithmetic case of rank-1, X admits a smooth toroidal compactification X¯ M obtained by adjoining an Abelian variety to each of its finitely many ends, and X¯ M can be blown down to a normal projective-algebraic variety X¯ Min by solving ∂¯ with L2-estimates with respect to the canonical Kähler-Einstein metric and by normalization. As an application, we give an alternative proof of results of Koziarz-Mok on the submersion problem in the case of complex-hyperbolic space forms of finite volume by adapting the cohomological arguments in the compact case to general hyperplane sections of the minimal projective-algebraic compactifications which avoid the isolated singularities.-
dc.languageengen_US
dc.publisherBirkhäuser Springeren_US
dc.relation.ispartofPerspectives in analysis, geometry, and topology: on the occasion of the 60th birthday of Oleg Viro-
dc.relation.ispartofseriesProgress in mathematics; v.296-
dc.titleProjective Algebraicity of Minimal Compactifications of Complex-Hyperbolic Space Forms of Finite Volumeen_US
dc.typeBook_Chapteren_US
dc.identifier.emailMok, N: nmok@hku.hken_US
dc.identifier.authorityMok, N=rp00763en_US
dc.identifier.doi10.1007/978-0-8176-8277-4_14-
dc.identifier.hkuros217113en_US
dc.identifier.spage331en_US
dc.identifier.epage354en_US
dc.identifier.isiWOS:000307264800015-
dc.publisher.placeNew Yorken_US

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