The Grauert--Grothendieck complex on differentiable spaces and a sheaf complex of Brylinski

TL;DR

This paper applies the Grauert--Grothendieck complex to differentiable spaces to analyze basic relative forms on inertia spaces, establishing a fine resolution of Brylinski's sheaf of functions.

Contribution

It demonstrates that the sheaf complex of basic relative forms provides a fine resolution of Brylinski's sheaf on inertia spaces, linking complex analysis with Lie group actions.

Findings

Sheaf complex of basic relative forms is a fine resolution of Brylinski's sheaf.

Provides new insights into the structure of inertia spaces under Lie group actions.

Connects complex geometry techniques with the study of differentiable spaces.

Abstract

We use the Grauert--Grothendieck complex on differentiable spaces to study basic relative forms on the inertia space of a compact Lie group action on a manifold. We prove that the sheaf complex of basic relative forms on the inertia space is a fine resolution of Bryliski's sheaf of functions on the inertia space.