Shimura varieties and hyperbolicity of moduli spaces

ac.commutative-algebra ag.algebraic-geometry cv.complex-variables nt.number-theory
Start Date
2018-05-28 
End Date
2018-06-01 
Institution
UQAM 
City
Montréal, QC 
Country
Canada 
Meeting Type
conference 
Homepage
https://www.i2m.univ-amu.fr/perso/erwan.rousseau/montreal/shimura.html 
Contact Name
 
Created
 
Modified
 

Description

The purpose is to gather experts from two mathematical communities, arithmetic geometry and complex geometry, who study from different perspectives Shimura varieties and questions related to hyperbolicity of moduli spaces.

The geometry of quotients of bounded symmetric domains $\Omega/\Gamma$ has been the subject of many works. Classical results (Mumford, Tai, Tsuyumine, etc.) and more recent ones (Bakker-Tsimerman, Brunebarbe, Cadorel, etc.) state that in most cases compactifications of such quotients have many differential forms. In particular, they are of general type. The geometry of entire curves has also been investigated (Nadel, Noguchi, Rousseau, etc.) showing that such quotients are in general hyperbolic modulo the boundary. Some recent works (Brunebarbe, Cadorel, etc.) have also established that all subvarieties of $\Omega/\Gamma$ are of general type. These results can be seen as illustrations in this context of the Green-Griffiths-Lang's conjectures which make a natural bridge with arithmetic problems.

Many conjectures on the arithmetic side indeed involve the study of subvarieties of Shimura varieties. There has been a lot of work around the so-called Andr\'e-Oort conjecture on special subvarieties of Shimura varieties (Klingler-Yafaev, Ullmo, Pila, Tsimerman, etc.) which was recently proved for $\mathcal{A}_g$, the moduli space of abelian varieties of dimension $g$, by J. Tsimerman by relying in particular on the so-called Colmez conjecture in average proved by Andreatta-Goren-Howard-Pera. There is on-going work using a similar line of attack that was successful to prove Andr\'e-Oort to investigate its generalization, labelled the Zilber-Pink conjecture for brevity, for example by Daw and Ren in the direction of the hyperbolic Ax-Schanuel conjecture. The interplay between complex geometry and arithmetic geometry relies on direct applications to Shimura varieties of general theorems from complex geometry; on the suggestive analogy between the complex category and the arithmetic setting; but also on more intertwined interactions.

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