Symmetries of Compact Riemann Surfaces

1. Auflage. 2010
ISBN: 978-3-642-14827-9
Verlag: Springer

Buch, Englisch, Band 2007, 164 Seiten, Format (B × H): 155 mm x 235 mm, Gewicht: 600 g

Reihe: Lecture Notes in Mathematics

The content of this monograph is situated in the intersection of important branches of mathematics like the theory of one complex variable, algebraic geometry, low dimensional topology and, from the point of view of the techniques used, com- natorial group theory. The main tool comes from the Uniformization Theorem for Riemannsurfaces,whichrelatesthetopologyofRiemannsurfacesandholomorphic or antiholomorphic actions on them to the algebra of classical cocompact Fuchsian groups or, more generally, non-euclidean crystallographic groups. Foundations of this relationship were established by A. M. Macbeath in the early sixties and dev- oped later by, among others, D. Singerman. Another important result in Riemann surface theory is the connection between Riemannsurfacesandtheir symmetrieswith complexalgebraiccurvesandtheirreal forms. Namely, there is a well known functorial bijective correspondence between compact Riemann surfaces and smooth, irreducible complex projective curves. The fact that a Riemann surface has a symmetry means, under this equivalence, that the corresponding complex algebraic curve has a real form, that is, it is the complex- cation of a real algebraic curve. Moreover, symmetries which are non-conjugate in the full group of automorphisms of the Riemann surface, correspond to real forms which are birationally non-isomorphic over the reals. Furthermore, the set of points xedbyasymmetryishomeomorphictoaprojectivesmoothmodeloftherealform.

Zielgruppe

Research

Autoren/Hrsg.

Bujalance, Emilio

Cirre, Francisco Javier

Gamboa, José Manuel

Gromadzki, Grzegorz

Fachgebiete

• Mathematik | Informatik Mathematik Mathematische Analysis Harmonische Analysis, Fourier-Mathematik
• Mathematik | Informatik Mathematik Topologie Mengentheoretische Topologie
• Mathematik | Informatik Mathematik Mathematische Analysis Funktionalanalysis

Inhaltsverzeichnis

Preliminaries.- On the Number of Conjugacy Classes of Symmetries of Riemann Surfaces.- Counting Ovals of Symmetries of Riemann Surfaces.- Symmetry Types of Some Families of Riemann Surfaces.- Symmetry Types of Riemann Surfaces with a Large Group of Automorphisms.