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Finiteness results for hyperbolic orbifold pairs

Laurine Weibel

Abstract

Noguchi proved that the set of dominant maps from a fixed variety to a fixed hyperbolic variety is finite. We extend this result to the setting of orbifold pairs, as introduced by Campana, under suitable assumptions. Certain compactness properties also allow us to prove that the set of orbifold pointed maps and the orbifold automorphism group are finite.

Finiteness results for hyperbolic orbifold pairs

Abstract

Noguchi proved that the set of dominant maps from a fixed variety to a fixed hyperbolic variety is finite. We extend this result to the setting of orbifold pairs, as introduced by Campana, under suitable assumptions. Certain compactness properties also allow us to prove that the set of orbifold pointed maps and the orbifold automorphism group are finite.

Paper Structure

This paper contains 24 sections, 39 theorems, 101 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Orbifolds
  3. First definitions
  4. Orbifold bundles
  5. Adapted coverings
  6. Orbifold (co)tangent bundle
  7. Positivity of orbifold cotangent bundle
  8. Orbifold hyperbolicity
  9. Definitions and first properties
  10. One dimensional case
  11. Hyperbolic line arrangment in P2
  12. Compactness result on the subset of surjective orbifold morphisms
  13. Reduction of the study
  14. The evaluation morphism is an orbifold morphism
  15. Finiteness result for hyperbolic uniformizable orbifold pairs with semi-negative orbifold canonical bundle
  16. ...and 9 more sections

Key Result

Theorem 1.1

Let $(X,\Delta_X)$ be a smooth projective orbifold and $(Y,\Delta_Y)$ be a hyperbolic smooth projective orbifold. If $(Y,\Delta_Y)$ is uniformizable or if $K_Y+\Delta_Y$ is pseudo-effective, then the set of surjective orbifold maps from $X$ on $(Y,\Delta_Y)$$\mathop{\mathrm{Sur}}\nolimits((X,\Delta_

Figures (4)

  • Figure 1: Quadrihedral
  • Figure 2: Blown up surface
  • Figure 3: Table of weights
  • Figure 4: Hyperbolic weights

Theorems & Definitions (89)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Proposition 1.6
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • ...and 79 more