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Uniformization of klt pairs by bounded symmetric domains

Patrick Graf, Aryaman Patel

Abstract

Given a complex-projective klt pair $(X, Δ)$ with standard coefficients and such that $K_X + Δ$ is ample, we determine necessary and sufficient conditions for the pair $(X, Δ)$ to be uniformized by a bounded symmetric domain. As an application, we obtain characterizations of orbifold quotients of the polydisc and of the four classical irreducible bounded symmetric domains in terms of Miyaoka-Yau-type Chern equalities.

Uniformization of klt pairs by bounded symmetric domains

Abstract

Given a complex-projective klt pair with standard coefficients and such that is ample, we determine necessary and sufficient conditions for the pair to be uniformized by a bounded symmetric domain. As an application, we obtain characterizations of orbifold quotients of the polydisc and of the four classical irreducible bounded symmetric domains in terms of Miyaoka-Yau-type Chern equalities.

Paper Structure

This paper contains 15 sections, 14 theorems, 66 equations.

Table of Contents

  1. Introduction
  2. Preliminaries on orbifolds
  3. Orbi-structures and developability
  4. Orbifolds as Deligne--Mumford stacks
  5. Principal bundles on orbifolds
  6. Hodge groups of Hermitian type
  7. Proof of \ref{['simpson orbifolds']}
  8. Proof of \ref{['main2']} and of the main result
  9. Chern classes of direct summands
  10. Extension form of a short exact sequence
  11. The Atiyah sequence
  12. Direct summands of the tangent bundle
  13. Proof of \ref{['c12 vanishing']}
  14. Quotients of the polydisc
  15. Quotients of the classical irreducible BSDs

Key Result

Theorem 1.2

Let $\mathcal{X} = (X, \Delta)$ be as in std. The following are equivalent:

Theorems & Definitions (37)

  • Theorem 1.2: Uniformization for klt pairs
  • Theorem 1.3: Criterion for quotient singularities
  • Theorem 1.4: Uniformization for orbifolds
  • Corollary 1.5: Uniformization by the polydisc
  • Corollary 1.6: Quotients of the Siegel upper half space, type C I
  • Corollary 1.7: Quotients of type D III
  • Remark 1.8
  • Corollary 1.9: Quotients of type A III
  • Remark 1.10
  • Remark 1.11
  • ...and 27 more