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Book Chapter: Projective Algebraicity of Minimal Compactifications of Complex-Hyperbolic Space Forms of Finite Volume
Title | Projective Algebraicity of Minimal Compactifications of Complex-Hyperbolic Space Forms of Finite Volume |
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Authors | |
Keywords | Hyperbolic space form Kähler–Einstein metric L2-method Minimal compactification |
Issue Date | 2012 |
Publisher | Birkhäuser 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? |
Abstract | Let 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 Identifier | http://hdl.handle.net/10722/187440 |
ISBN | |
ISSN | 2020 SCImago Journal Rankings: 0.870 |
ISI Accession Number ID | |
Series/Report no. | Progress in mathematics; v.296 |
DC Field | Value | Language |
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dc.contributor.author | Mok, N | en_US |
dc.date.accessioned | 2013-08-20T12:45:18Z | - |
dc.date.available | 2013-08-20T12:45:18Z | - |
dc.date.issued | 2012 | en_US |
dc.identifier.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 | en_US |
dc.identifier.isbn | 9780817682767 | - |
dc.identifier.issn | 0743-1643 | - |
dc.identifier.uri | http://hdl.handle.net/10722/187440 | - |
dc.description.abstract | Let 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.language | eng | en_US |
dc.publisher | Birkhäuser Springer | en_US |
dc.relation.ispartof | Perspectives in analysis, geometry, and topology: on the occasion of the 60th birthday of Oleg Viro | - |
dc.relation.ispartofseries | Progress in mathematics; v.296 | - |
dc.subject | Hyperbolic space form | - |
dc.subject | Kähler–Einstein metric | - |
dc.subject | L2-method | - |
dc.subject | Minimal compactification | - |
dc.title | Projective Algebraicity of Minimal Compactifications of Complex-Hyperbolic Space Forms of Finite Volume | en_US |
dc.type | Book_Chapter | en_US |
dc.identifier.email | Mok, N: nmok@hku.hk | en_US |
dc.identifier.authority | Mok, N=rp00763 | en_US |
dc.identifier.doi | 10.1007/978-0-8176-8277-4_14 | - |
dc.identifier.scopus | eid_2-s2.0-84874118258 | - |
dc.identifier.hkuros | 217113 | en_US |
dc.identifier.spage | 331 | en_US |
dc.identifier.epage | 354 | en_US |
dc.identifier.eissn | 2296-505X | - |
dc.identifier.isi | WOS:000307264800015 | - |
dc.publisher.place | New York | en_US |
dc.identifier.issnl | 0743-1643 | - |