On the existence of logarithmic and orbifold jet differentials

TL;DR

This paper develops a framework for orbifold jet differentials on directed orbifolds, providing conditions for their existence, which advances understanding of algebraic differential operators in orbifold geometry.

Contribution

It introduces the concept of directed orbifolds and constructs an algebra of orbifold jet differentials with curvature estimates, offering new tools for studying differential operators on orbifolds.

Findings

Established algebraic structures for orbifold jet differentials.

Derived curvature estimates for the directed orbifold structure.

Provided effective conditions for the existence of global orbifold jet differentials.

Abstract

We introduce the concept of directed orbifold, namely triples (X, V, D) formed by a directed algebraic or analytic variety (X, V), and a ramification divisor D, where V is a coherent subsheaf of the tangent bundle TX. In this context, we introduce an algebra of orbifold jet differentials and their sections. These jet sections can be seen as algebraic differential operators acting on germs of curves, with meromorphic coefficients, whose poles are supported by D and multiplicities are bounded by the ramification indices of the components of D. We estimate precisely the curvature tensor of the corresponding directed structure V[D] in the general orbifold case-with a special attention to the compact case D = 0 and to the logarithmic situation where the ramification indices are infinite. Using holomorphic Morse inequalities on the tautological line bundle of the projectivized orbifold Green-Griffiths bundle, we finally obtain effective sufficient conditions for the existence of global orbifold jet differentials.