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Geometriae Dedicata

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

Singularities of ball quotients

  • Niko Behrens
Original Paper
  • 65 Downloads

Abstract

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.

Keywords

Ball quotients Canonical singularities Toroidal compactification 

List of symbols

\({\left(\frac{\cdot}{\cdot}\right)}\)

Kronecker symbol

(a, b)

Greatest common divisor for integers a, b

\({\varphi}\)

Euler’s (number theoretic) phi function

\({\phi_r}\)

rth cyclotomic polynomial

{q}

Fractional part of a rational number q

HA

Complex conjugate transpose of a complex matrix A

\({\mathbb{C}{H^n}}\)

n-dimensional complex hyperbolic space

\({\mathbb{P}^n_\mathbb{C}}\)

n-dimensional complex projective space

\({B^n_\mathbb{C}}\)

n-dimensional complex ball

U(Λ)

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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