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

TL;DR

This paper proves that Green--Griffiths--Demailly hyperbolicity thresholds are invariant under passing from complex manifolds to orbifolds with the same Kähler class, showing their fundamental geometric nature.

Contribution

It establishes the structural invariance of GGD thresholds for orbifolds and provides an orbifold Riemann--Roch formula demonstrating the independence from orbifold singularities.

Findings

GGD thresholds are unchanged when passing to orbifolds with the same Kähler class.

The orbifold Riemann--Roch formula shows only the identity sector affects the leading term.

Invariant jet differential existence depends solely on the coarse Kähler class.

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 K\"ahler 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 $\chi$, 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 K\"ahler class--hence orbifold compactification or rigidification does not alter the GGD threshold or the hyperbolicity locus.