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Orbifold Classifying Spaces and Quotients of complex Tori

Fabrizio Catanese

Abstract

In this paper we characterize the quotients $ X = T/G$ of a complex torus $T$ by the action of a finite group $G$ as the Kähler orbifold classifying spaces of the even Euclidean cristallographic groups $Γ$, and we prove other similar and stronger characterizations.

Orbifold Classifying Spaces and Quotients of complex Tori

Abstract

In this paper we characterize the quotients of a complex torus by the action of a finite group as the Kähler orbifold classifying spaces of the even Euclidean cristallographic groups , and we prove other similar and stronger characterizations.
Paper Structure (8 sections, 6 theorems, 16 equations)

This paper contains 8 sections, 6 theorems, 16 equations.

Table of Contents

  1. Complex orbifolds, Deligne-Mostow orbifolds, orbifold fundamental groups, orbifold coverings
  2. Euclidean cristallographic groups and Actions of a finite group $G$ on a complex torus $T$
  3. Proof of the Main Theorems
  4. Properties of $X=T/G$ and proof of the easier implications
  5. The Kähler property
  6. Proof of Theorem \ref{['proj']}.
  7. Proof of Theorems \ref{['kahler']} and \ref{['3']}
  8. Parametrizing Families

Key Result

Theorem 1

Finite quotients of complex Abelian varieties are: (i) the Deligne-Mostow projective orbifolds which are orbifold classifying spaces for even Euclidean cristallographic groups $\Gamma$, or more generally (ii) the complex projective orbifolds with KLT singularities which are orbifold classifying spac

Theorems & Definitions (14)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Definition 4
  • Remark 5
  • Definition 6
  • Remark 7
  • Definition 8
  • Theorem 9
  • Proposition 10
  • ...and 4 more