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Conference Paper: On the Zariski closure of an infinite number of totally geodesic subvarieties of Ω/Γ
Title  On the Zariski closure of an infinite number of totally geodesic subvarieties of Ω/Γ 

Other Titles  On the Zariski closure of an infinite number of totally geodesic subvarieties of Omega/Gamma 
Authors  
Issue Date  2013 
Citation  The Abel Symposium 2013: Complex geometry, Trondheim, Norway, 25 July 2013 How to Cite? 
Abstract  Let $Omega$ be a bounded symmetric domain, $Gamma subset ext{Aut}(Omega)$ be a torsionfree lattice, $X := Omega/Gamma$. Let $Z subset X$ be an irreducible quasiprojective variety such that $Z$ is the Zariski closure of the union of an infinite family of totallygeodesic complex subvarieties $S_alpha subset Z, , alpha in A$. Under a nondegeneracy condition one expects $Z$ to be also totally geodesic, so that $Z$ is again uniformized by a bounded symmetric domain. This setup is related to a wellknown problem on Shimura varieties $X = Omega/Gamma$ in which one tries to characterize the Zariski closure of an infinite number of `special' subvarieties $S_alpha$. Here special subvarieties are defined by arithmetic conditions, but it is known that they are always totally geodesic. While the case where $S_alpha$ are 0dimensional, for which the problem is called the AndréOort Conjecture, cannot be dealt with directly using methods of complex geometry, the case where $S_alpha$ are of positive dimension is naturally a problem in complex geometry. From our complexanalytic perspective, no arithmeticity assumption is placed on $Gamma subset ext{Aut}(Omega)$, and the `distinguished' subvarieties are simply the totallygeodesic subvarieties of $X$.
Using methods of Kähler geometry, we solve the aforementioned problem in the rank1 case. A generalization of the argument to bounded symmetric domains $Omega$ leads to the study of holomorphic isometries from complex unit balls $B^m$ to $Omega$. We explain a method along this line of thoughts for solving the general problem, and illustrate how the problem is solved when $Z$ is a complex surface and $S_alpha subset Z$ are totallygeodesic holomorphic curves using our recent results on holomorphic isometries with respect to the Bergman metric. 
Description  Plenary Lecture 
Persistent Identifier  http://hdl.handle.net/10722/269936 
DC Field  Value  Language 

dc.contributor.author  Mok, N   
dc.date.accessioned  20190516T04:12:24Z   
dc.date.available  20190516T04:12:24Z   
dc.date.issued  2013   
dc.identifier.citation  The Abel Symposium 2013: Complex geometry, Trondheim, Norway, 25 July 2013   
dc.identifier.uri  http://hdl.handle.net/10722/269936   
dc.description  Plenary Lecture   
dc.description.abstract  Let $Omega$ be a bounded symmetric domain, $Gamma subset ext{Aut}(Omega)$ be a torsionfree lattice, $X := Omega/Gamma$. Let $Z subset X$ be an irreducible quasiprojective variety such that $Z$ is the Zariski closure of the union of an infinite family of totallygeodesic complex subvarieties $S_alpha subset Z, , alpha in A$. Under a nondegeneracy condition one expects $Z$ to be also totally geodesic, so that $Z$ is again uniformized by a bounded symmetric domain. This setup is related to a wellknown problem on Shimura varieties $X = Omega/Gamma$ in which one tries to characterize the Zariski closure of an infinite number of `special' subvarieties $S_alpha$. Here special subvarieties are defined by arithmetic conditions, but it is known that they are always totally geodesic. While the case where $S_alpha$ are 0dimensional, for which the problem is called the AndréOort Conjecture, cannot be dealt with directly using methods of complex geometry, the case where $S_alpha$ are of positive dimension is naturally a problem in complex geometry. From our complexanalytic perspective, no arithmeticity assumption is placed on $Gamma subset ext{Aut}(Omega)$, and the `distinguished' subvarieties are simply the totallygeodesic subvarieties of $X$. Using methods of Kähler geometry, we solve the aforementioned problem in the rank1 case. A generalization of the argument to bounded symmetric domains $Omega$ leads to the study of holomorphic isometries from complex unit balls $B^m$ to $Omega$. We explain a method along this line of thoughts for solving the general problem, and illustrate how the problem is solved when $Z$ is a complex surface and $S_alpha subset Z$ are totallygeodesic holomorphic curves using our recent results on holomorphic isometries with respect to the Bergman metric.   
dc.language  eng   
dc.relation.ispartof  The Abel Symposium   
dc.title  On the Zariski closure of an infinite number of totally geodesic subvarieties of Ω/Γ   
dc.title.alternative  On the Zariski closure of an infinite number of totally geodesic subvarieties of Omega/Gamma   
dc.type  Conference_Paper   
dc.identifier.email  Mok, N: nmok@hku.hk   
dc.identifier.authority  Mok, N=rp00763   
dc.identifier.hkuros  217115   