Families of Riemann surfaces, uniformization and arithmeticity
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- by Gabino González-Diez and Sebastián Reyes-Carocca PDF
- Trans. Amer. Math. Soc. 370 (2018), 1529-1549 Request permission
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
- MR Author ID: 293212
- Email: gabino.gonzalez@uam.es
- Sebastián Reyes-Carocca
- Affiliation: Departamento de Matemática y Estadística, Universidad de La Frontera, Temuco, Chile
- MR Author ID: 981144
- ORCID: 0000-0001-7832-4150
- Email: sebastian.reyes@ufrontera.cl
- 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 - © Copyright 2017 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 370 (2018), 1529-1549
- MSC (2010): Primary 32G15, 14J20, 14J29
- DOI: https://doi.org/10.1090/tran/6988
- MathSciNet review: 3739184