Desingularizing compact Lie group actions

TL;DR

This paper reviews the structure of G-manifolds under compact Lie group actions and introduces a method to desingularize these actions, enabling the transfer of analysis and geometric results from simplified models to original manifolds.

Contribution

It presents a novel approach to reduce singular strata in G-manifolds and constructs equivariant Dirac-type operators on these desingularized spaces.

Findings

Reduction of singular strata in G-manifolds.

Construction of equivariant Dirac operators on desingularized manifolds.

Translation of analysis results from simplified models to original manifolds.

Abstract

This note surveys the well-known structure of G-manifolds and summarizes parts of two papers that have not yet appeared in print: one with joint with J. Bruning and F. W. Kamber, and another with I. Prokhorenkov. In particular, from a given manifold on which a compact Lie group acts smoothly, we construct a sequence of manifolds on which the same Lie group acts, but with fewer levels of singular strata. Global analysis and geometric results on the simpler manifolds may be translated to results on the original manifold. Further, we show that by utilizing bundles over a G-manifold with singular strata, we may construct natural equivariant transverse Dirac-type operators that have properties similar to Dirac operators on closed manifolds.