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Families of Riemann surfaces, uniformization and arithmeticity

Authors: Gabino González-Diez and Sebastián Reyes-Carocca
Journal: Trans. Amer. Math. Soc. 370 (2018), 1529-1549
MSC (2010): Primary 32G15, 14J20, 14J29
Published electronically: November 7, 2017
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Abstract: A consequence of the results of Bers and Griffiths on the uniformization of complex algebraic varieties is that the universal cover of a family of Riemann surfaces, with base and fibers of finite hyperbolic type, is a contractible $ 2-$dimensional domain that can be realized as the graph of a holomorphic motion of the unit disk.

In this paper we determine which holomorphic motions give rise to these uniformizing domains and characterize which among them correspond to arithmetic families (i.e. families defined over number fields). Then we apply these results to characterize the arithmeticity of complex surfaces of general type in terms of the biholomorphism class of the $ 2-$dimensional domains that arise as universal covers of their Zariski open subsets. For the important class of Kodaira fibrations this criterion implies that arithmeticity can be read from the universal cover. All this is very much in contrast with the corresponding situation in complex dimension one, where the universal cover is always the unit disk.

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Additional Information

Gabino González-Diez
Affiliation: Departamento de Matemáticas, Universidad Autónoma de Madrid, Madrid, Spain

Sebastián Reyes-Carocca
Affiliation: Departamento de Matemática y Estadística, Universidad de La Frontera, Temuco, Chile

Keywords: Holomorphic families of Riemann surfaces, complex surfaces and their universal covers, fields of definition, holomorphic motions
Received by editor(s): October 29, 2015
Received by editor(s) in revised form: May 13, 2016
Published electronically: November 7, 2017
Additional Notes: Both authors were partially supported by Spanish MEyC Grant MTM 2012-31973
The second author was also partially supported by Becas Chile, Universidad de La Frontera, Fondecyt Postdoctoral Project 3160002 and Project Anillo ACT1415 PIA CONICYT
Article copyright: © Copyright 2017 American Mathematical Society

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