Differential forms on orbifolds with corners

TL;DR

This paper develops a comprehensive framework for differential forms and currents on orbifolds with corners, emphasizing formalism that avoids orbifold charts and proves independence of representations.

Contribution

It introduces a formalism for differential forms on orbifolds with corners using groupoids, ensuring independence from specific groupoid choices.

Findings

Differential forms form a Fréchet space independent of groupoid representation.

Currents' dual space is well-defined and independent of the orbifold presentation.

Formalism simplifies operations like pull-back and push-forward on orbifolds with corners.

Abstract

Motivated by symplectic geometry, we give a detailed account of differential forms and currents on orbifolds with corners, the pull-back and push-forward operations, and their fundamental properties. We work within the formalism where the category of orbifolds with corners is obtained as a localization of the category of \'etale proper groupoids with corners. Constructions and proofs are formulated in terms of the structure maps of the groupoids, avoiding the use of orbifold charts. The Fr\'echet space of differential forms on an orbifold and the dual space of currents are shown to be independent of which \'etale proper groupoid is chosen to represent the orbifold.