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Orbifolds and their uniformization. (English) Zbl 1126.32020
Holzapfel, Rolf-Peter (ed.) et al., Arithmetic and geometry around hypergeometric functions. Lecture notes of a CIMPA summer school held at Galatasaray University, Istanbul, Turkey, June 13–25, 2005. Basel: Birkhäuser (ISBN 978-3-7643-8283-4/hbk). Progress in Mathematics 260, 373-406 (2007).
These notes are based on talks given by the author at the CIMPA summer school [Arithmetic and Geometry around Hypergeometric Functions. Lecture notes of a CIMPA summer school held at Galatasaray University, Istanbul, Turkey, June 13–25, Progress in Mathematics 260. Basel: Birkhäuser (2007; Zbl 1109.14002)] held in Istanbul and the EMS summer school “Braid Groups and Related Topics (2005)” held in Tiberias.
An orbifold is a space locally modeled on a smooth manifold modulo a finite group action, which is said to be uniformizable if it is a global quotient. Orbifolds appear natually in various fields of mathematics and physics. The author focuses on the uniformization problem and considers almost exclusively orbifolds with a smooth base space. In the first section he presents some background on branched coverings. Next, he presents facts and definitions about orbifolds for which he studies the local structure and singularities, especially in dimension 2. In section 4 he sketches the solutions of the global uniformization problem for some special orbifolds. In particular, he presents a complete classification of Abelian finite smooth branched coverings of $$\mathbb{P}^2$$. There are also many examples of non-Abelian coverings.
For the entire collection see [Zbl 1109.14002].

##### MSC:
 32Q30 Uniformization of complex manifolds 14J25 Special surfaces
##### Keywords:
orbifold; orbiface; uniformization; ball-quotient
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