Belyi's Theorem for Complex Surfaces

Belyi's theorem states that a compact Riemann surface C can be defined over a number field if and only if there is on it a meromorphic function f with three critical values. Such functions (resp. Riemann surfaces) are called Belyi functions (resp. Belyi surfaces). Alternatively Belyi surfaces c...

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Bibliographic Details
Published in:American journal of mathematics Vol. 130; no. 1; pp. 59 - 74
Main Author: González-Diez, Gabino
Format: Journal Article
Language:English
Published: Baltimore, MD Johns Hopkins University Press 01.02.2008
Subjects:
Algebra
Algebraic geometry
Critical points
Critical values
Exact sciences and technology
Functions of a complex variable
General mathematics
General, history and biography
Mathematical analysis
Mathematical surfaces
Mathematical theorems
Mathematics
Minimal surfaces
Morphisms
Pencils
Riemann surfaces
Sciences and techniques of general use
Several complex variables and analytic spaces
Modular group
Critical value
Riemann function
Number field
Subgroup
Two-dimensional calculations
Mathematics
Free group
Riemann surface
Torsion
Meromorphic function
Classical group
ISSN:0002-9327, 1080-6377, 1080-6377
Online Access:Get full text
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Summary:Belyi's theorem states that a compact Riemann surface C can be defined over a number field if and only if there is on it a meromorphic function f with three critical values. Such functions (resp. Riemann surfaces) are called Belyi functions (resp. Belyi surfaces). Alternatively Belyi surfaces can be characterized as those which contain a proper Zariski open subset uniformised by a torsion free subgroup of the classical modular group PSL₂(Z). In this article we establish a result analogous to Belyi's theorem in complex dimension two. It turns out that the role of Belyi functions is now played by (composed) Lefschetz pencils with three critical values while the analogous to torsion free subgroups of the modular group will be certain extensions of them acting on a Bergmann domain of C². These groups were first introduced by Bers and Griffiths.
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ISSN:0002-9327
1080-6377
1080-6377
DOI:10.1353/ajm.2008.0004