Geometriae Dedicata

, Volume 159, Issue 1, pp 389–407 | Cite as

Singularities of ball quotients

  • Niko Behrens
Original Paper


We prove a result on the singularities of ball quotients \({\Gamma\backslash\mathbb{C}{H^n}}\) by an arithmetic group. More precisely, we show that a ball quotient has at most canonical singularities under certain restrictions on the dimension n and the underlying lattice. We also extend this result to the toroidal compactification.


Ball quotients Canonical singularities Toroidal compactification 

List of symbols


Kronecker symbol

(a, b)

Greatest common divisor for integers a, b


Euler’s (number theoretic) phi function


rth cyclotomic polynomial


Fractional part of a rational number q


Complex conjugate transpose of a complex matrix A


n-dimensional complex hyperbolic space


n-dimensional complex projective space


n-dimensional complex ball


Automorphism group of lattice Λ

U(n, 1)

Unitary group of signature (n, 1)

Mathematics Subject Classification (2000)

14E15 14J17 14L30 14M17 14M27 


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Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  1. 1.Institut für Algebraische Geometrie, Leibniz Universität HannoverHannoverGermany

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