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Conference Paper: Uniformization Problems on Subvarieties of Finite-Volume Quotient Spaces of Bounded Symmetric Domains
Title | Uniformization Problems on Subvarieties of Finite-Volume Quotient Spaces of Bounded Symmetric Domains |
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Authors | |
Issue Date | 2019 |
Citation | The 8th International Congress of Chinese Mathematicians (ICCM 2019), Tsinghua University, Beijing, China, 9-14 June 2019 How to Cite? |
Abstract | By the Uniformization Theorem a compact Riemann surface of genus ≥ 2 is uniformized by the unit disk Δ and equivalently by the upper half plane H. H is also the universal covering space of the moduli space of elliptic curves equipped with an appropriate level structure. In Several Complex Variables, the Siegel upper half plane Hg; g≥ 1, is an analogue of H = H1, and it is the universal covering space of moduli spaces of polarized Abelian varieties with appropriate level structures. Hg belongs, up to biholomorphic equivalence, to the set of bounded symmetric domains, on which a great deal of mathematical research is taking place. Especially, finite-volume quotients of bounded symmetric domains Ω, which are naturally quasi-projective varieties, are objects of immense interest to Several Complex Variables, Algebraic Geometry, Arithmetic Geometry and Number Theory, and an important topic is the study of covering spaces of algebraic subsets of such quasi-projective varieties. While a lot has already been achieved in the case of Shimura varieties by means of methods of Diophantine Geometry, Model Theory, Hodge Theory and Complex Differential Geometry, techniques for the general case of not necessarily arithmetic quotients Ω/T, which are necessarily quasi-projective by earlier works of the speaker with Jiaqing Zhong. Especially, we will explain how the study of holomorphic isometric embeddings of the Poincaré disk and more generally complex unit balls into bounded symmetric domains can be further developed to derive uniformization theorems for bi-algebraic varieties and more generally for the Zariski closure of images of algebraic sets under the universal covering map. |
Description | Jointly hosted by Yau Mathematical Sciences Center (YMSC) and Department of Mathematical Sciences of Tsinghua University |
Persistent Identifier | http://hdl.handle.net/10722/271510 |
DC Field | Value | Language |
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dc.contributor.author | Mok, N | - |
dc.date.accessioned | 2019-07-02T09:17:13Z | - |
dc.date.available | 2019-07-02T09:17:13Z | - |
dc.date.issued | 2019 | - |
dc.identifier.citation | The 8th International Congress of Chinese Mathematicians (ICCM 2019), Tsinghua University, Beijing, China, 9-14 June 2019 | - |
dc.identifier.uri | http://hdl.handle.net/10722/271510 | - |
dc.description | Jointly hosted by Yau Mathematical Sciences Center (YMSC) and Department of Mathematical Sciences of Tsinghua University | - |
dc.description.abstract | By the Uniformization Theorem a compact Riemann surface of genus ≥ 2 is uniformized by the unit disk Δ and equivalently by the upper half plane H. H is also the universal covering space of the moduli space of elliptic curves equipped with an appropriate level structure. In Several Complex Variables, the Siegel upper half plane Hg; g≥ 1, is an analogue of H = H1, and it is the universal covering space of moduli spaces of polarized Abelian varieties with appropriate level structures. Hg belongs, up to biholomorphic equivalence, to the set of bounded symmetric domains, on which a great deal of mathematical research is taking place. Especially, finite-volume quotients of bounded symmetric domains Ω, which are naturally quasi-projective varieties, are objects of immense interest to Several Complex Variables, Algebraic Geometry, Arithmetic Geometry and Number Theory, and an important topic is the study of covering spaces of algebraic subsets of such quasi-projective varieties. While a lot has already been achieved in the case of Shimura varieties by means of methods of Diophantine Geometry, Model Theory, Hodge Theory and Complex Differential Geometry, techniques for the general case of not necessarily arithmetic quotients Ω/T, which are necessarily quasi-projective by earlier works of the speaker with Jiaqing Zhong. Especially, we will explain how the study of holomorphic isometric embeddings of the Poincaré disk and more generally complex unit balls into bounded symmetric domains can be further developed to derive uniformization theorems for bi-algebraic varieties and more generally for the Zariski closure of images of algebraic sets under the universal covering map. | - |
dc.language | eng | - |
dc.relation.ispartof | International Congress of Chinese Mathematicians (ICCM), 2019 | - |
dc.title | Uniformization Problems on Subvarieties of Finite-Volume Quotient Spaces of Bounded Symmetric Domains | - |
dc.type | Conference_Paper | - |
dc.identifier.email | Mok, N: nmok@hku.hk | - |
dc.identifier.authority | Mok, N=rp00763 | - |
dc.identifier.hkuros | 298200 | - |