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

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.