Belyi's theorem for complex surfaces

• Autores: Gabino González Díez
• Localización: American journal of mathematics, ISSN 0002-9327, Vol. 130, Nº 1, 2008, págs. 59-74
• Idioma: inglés
• Texto completo no disponible (Saber más ...)
• Resumen

  • 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 SL2(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 2. These groups were first introduced by Bers and Griffiths.