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Mok tensors and Orbifold Quotients of Bounded symmetric domains without ball factors

Fabrizio Catanese, Marco Franciosi

TL;DR

This work characterizes compact orbifolds ${\mathcal X}=(X,D)$ arising as quotients of a bounded symmetric domain ${\mathcal D}$ by a properly discontinuous group, under the no-ball-factor hypothesis, by three intertwined criteria: the orbifold canonical divisor $K_{\mathcal X}=K_X+D$ is ample, ${\mathcal X}$ carries a Mok curvature-type tensor of orbifold type, and the orbifold fundamental group has a torsion-free finite index subgroup yielding a reduced smooth orbifold cover. The authors develop a descent theory for curvature-type tensors to orbifold settings, and show that the existence of a Mok tensor forces a holonomy decomposition corresponding to a product of irreducible bounded symmetric domains not containing balls. They then prove both necessary and sufficient conditions for uniformization, with refinements in the Deligne–Mostow and klt (Kawamata log terminal) cases, including a smoothness conclusion via the Lipman–Zariski framework in high dimension. Their results extend prior tube-type domain descriptions to the broader class of domains without ball factors, connecting complex-analytic, differential-geometric, and algebraic techniques to achieve a complete classification up to finite covers. The work provides a robust bridge between orbifold geometry, Mok’s tensors, and the structure of bounded symmetric domain quotients, with potential implications for moduli theory and the study of orbifold uniformization.

Abstract

In this paper we characterize the compact orbifolds, quotients $ X = \mathcal{D}/ Γ$ of a bounded symmetric domain $\mathcal{D}$ with no higher dimensional ball factor by the action of a discontinuous group $Γ$, as those projective orbifolds with ample orbifold canonical divisor which admit a Mok curvature type tensor of orbifold type and satisfying certain other conditions implying the existence of a finite smooth covering.

Mok tensors and Orbifold Quotients of Bounded symmetric domains without ball factors

TL;DR

This work characterizes compact orbifolds arising as quotients of a bounded symmetric domain by a properly discontinuous group, under the no-ball-factor hypothesis, by three intertwined criteria: the orbifold canonical divisor is ample, carries a Mok curvature-type tensor of orbifold type, and the orbifold fundamental group has a torsion-free finite index subgroup yielding a reduced smooth orbifold cover. The authors develop a descent theory for curvature-type tensors to orbifold settings, and show that the existence of a Mok tensor forces a holonomy decomposition corresponding to a product of irreducible bounded symmetric domains not containing balls. They then prove both necessary and sufficient conditions for uniformization, with refinements in the Deligne–Mostow and klt (Kawamata log terminal) cases, including a smoothness conclusion via the Lipman–Zariski framework in high dimension. Their results extend prior tube-type domain descriptions to the broader class of domains without ball factors, connecting complex-analytic, differential-geometric, and algebraic techniques to achieve a complete classification up to finite covers. The work provides a robust bridge between orbifold geometry, Mok’s tensors, and the structure of bounded symmetric domain quotients, with potential implications for moduli theory and the study of orbifold uniformization.

Abstract

In this paper we characterize the compact orbifolds, quotients of a bounded symmetric domain with no higher dimensional ball factor by the action of a discontinuous group , as those projective orbifolds with ample orbifold canonical divisor which admit a Mok curvature type tensor of orbifold type and satisfying certain other conditions implying the existence of a finite smooth covering.
Paper Structure (14 sections, 8 theorems, 39 equations)

This paper contains 14 sections, 8 theorems, 39 equations.

Table of Contents

  1. Complex orbifolds, Deligne-Mostow orbifolds, orbifold fundamental groups, orbifold coverings
  2. Notation
  3. Orbifold coverings
  4. Local orbifold fundamental group
  5. Algebraic curvature-type tensors, their First Mok characteristic varieties, and Mok tensors
  6. Symmetric domains and descent of curvature type tensors
  7. Curvature type tensors and their descent
  8. Proof of the main Theorems
  9. Necessary conditions.
  10. Sufficient conditions.
  11. Proof of Theorem \ref{['easier']}
  12. Proof of Theorem \ref{['dm']}
  13. Proof of Theorem \ref{['dm2']}.
  14. Proof of Theorem \ref{['theorem klt']}.

Key Result

Theorem 1

The global compact complex orbifolds ${\mathcal{X}}=(X,D)$ of bounded symmetric domains ${\mathcal{D}}$ with no irreducible factors isomorphic to a higher dimensional ball are the projective complex orbifolds such that: (1) $K_{{\mathcal{X}}} := K_X + D= K_X + \sum_i \frac{m_i-1}{m_i} D_i$ is ample which is of orbifold type (see Def. orbifold type and Lemma descent); (3) the orbifold fundamental

Theorems & Definitions (17)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Definition 5: Orbifold covering (see for instance d-m)
  • Definition 6
  • Proposition 7
  • proof
  • Definition 8
  • Remark 9
  • ...and 7 more