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Structural Invariance of Green--Griffiths--Demailly Thresholds on Compact Complex Orbifolds

Gunhee Cho, Myungsin Cho

TL;DR

The paper addresses whether Green--Griffiths--Demailly hyperbolicity thresholds are preserved when moving from a smooth compact complex manifold to a compact complex orbifold or analytic Deligne--Mumford stack with the same coarse Kähler class. It develops a chartwise orbifold HRR framework and shows that the leading term in the Euler characteristic of invariant jet bundles comes from the identity sector and scales by the generic stabilizer factor $1/s$, while twisted sectors contribute only $O(m^{n-1})$ due to fixed-point denominators. By constructing and descending the Demailly--Semple tower on orbifolds, and proving L^2 vanishing and Bochner-type estimates in the orbifold setting, the authors show that the existence range of invariant jet differentials—and hence the GGD threshold—depends only on the coarse Kähler class. Consequently, orbifold compactification or rigidification does not alter the GGD thresholds, with the results tying together Satake–Kawasaki index theory and stack Riemann--Roch to establish a robust invariance principle with potential applications to moduli and hyperbolicity phenomena.

Abstract

We prove that the Green--Griffiths--Demailly (GGD) hyperbolicity thresholds are structurally invariant. In other words, the minimal jet order and asymptotic growth rate at which invariant jet differentials appear remain unchanged when passing from a compact complex manifold to any compact smooth analytic Deligne--Mumford stack (orbifold) with the same coarse Kähler class. We establish an orbifold Riemann--Roch formula showing that only the identity sector contributes to the leading $m^n$ term of the Euler characteristic $χ$, while all twisted sectors contribute only $O(m^{n-1})$. Together with curvature--positivity properties of the Demailly--Semple tower, this implies that the existence range of invariant jet differentials depends solely on the coarse Kähler class--hence orbifold compactification or rigidification does not alter the GGD threshold or the hyperbolicity locus.

Structural Invariance of Green--Griffiths--Demailly Thresholds on Compact Complex Orbifolds

TL;DR

The paper addresses whether Green--Griffiths--Demailly hyperbolicity thresholds are preserved when moving from a smooth compact complex manifold to a compact complex orbifold or analytic Deligne--Mumford stack with the same coarse Kähler class. It develops a chartwise orbifold HRR framework and shows that the leading term in the Euler characteristic of invariant jet bundles comes from the identity sector and scales by the generic stabilizer factor , while twisted sectors contribute only due to fixed-point denominators. By constructing and descending the Demailly--Semple tower on orbifolds, and proving L^2 vanishing and Bochner-type estimates in the orbifold setting, the authors show that the existence range of invariant jet differentials—and hence the GGD threshold—depends only on the coarse Kähler class. Consequently, orbifold compactification or rigidification does not alter the GGD thresholds, with the results tying together Satake–Kawasaki index theory and stack Riemann--Roch to establish a robust invariance principle with potential applications to moduli and hyperbolicity phenomena.

Abstract

We prove that the Green--Griffiths--Demailly (GGD) hyperbolicity thresholds are structurally invariant. In other words, the minimal jet order and asymptotic growth rate at which invariant jet differentials appear remain unchanged when passing from a compact complex manifold to any compact smooth analytic Deligne--Mumford stack (orbifold) with the same coarse Kähler class. We establish an orbifold Riemann--Roch formula showing that only the identity sector contributes to the leading term of the Euler characteristic , while all twisted sectors contribute only . Together with curvature--positivity properties of the Demailly--Semple tower, this implies that the existence range of invariant jet differentials depends solely on the coarse Kähler class--hence orbifold compactification or rigidification does not alter the GGD threshold or the hyperbolicity locus.

Paper Structure

This paper contains 25 sections, 44 theorems, 98 equations.

Table of Contents

  1. Introduction
  2. Orbifold Hirzebruch--Riemann--Roch in the Chartwise (Satake) Formalism and Application to the Demailly--Semple Tower
  3. From analytic Deligne-Mumford stacks to complex orbifolds (rigidification and effectiveness)
  4. Hirzebruch--Riemann--Roch and its orbifold variants
  5. Orbifold Riemann--Roch: chartwise fixed-point formula
  6. Demailly--Semple tower on orbifolds: construction and descent
  7. Asymptotic packages for positivity and vanishing
  8. Green--Griffiths--Demailly Degeneracy on Analytic DM Stacks
  9. Notation and standing conventions
  10. Atlases, groupoids, and jets
  11. Curvature–negativity–positivity package
  12. (i) Coarse positivity.
  13. (ii) Vertical negativity.
  14. (iii) Semple positivity.
  15. (iv) Log completeness.
  16. ...and 10 more sections

Key Result

Theorem 1

Let $\pi\colon \mathcal{X}\to Y$ be the coarse moduli map of a compact smooth analytic Deligne--Mumford stack (orbifold) $\mathcal{X}$, and let $A$ be an ample line bundle on $Y$ endowed with a smooth positively curved metric. Set $L=\pi^*A$. Then there exist integers $m_0,q_0>0$ such that for all $ Thus, orbifold compactification and rigidification neither alter nor shift the GGD threshold.

Theorems & Definitions (100)

  • Theorem : Structural invariance of GGD thresholds [Theorem. \ref{['thm:main-threshold']}]
  • Definition 1
  • Definition 2: Effectivity and compactness for analytic DM stacks
  • Lemma 1: Local finite-quotient presentation
  • proof
  • Lemma 2: Rigidification of ineffectivity
  • proof
  • Proposition 1: Stack–orbifold correspondence up to rigidification
  • proof
  • Proposition 2: Compactness criteria and coarse moduli
  • ...and 90 more