Holonomy of parabolic geometries near isolated higher-order fixed points

TL;DR

This paper investigates the holonomy properties of parabolic geometries near isolated higher-order fixed points, establishing conditions for trivial holonomy and applying results to classify certain geometric structures with automorphisms.

Contribution

It introduces a general criterion based on isotropy behavior for the existence of open subsets with trivial holonomy in parabolic geometries, independent of curvature assumptions.

Findings

Characterized all almost c-projective structures with higher-order fixed points.

Classified nondegenerate partially integrable almost CR structures with fixed points.

Established new relations between holonomy limits and automorphism fixed points.

Abstract

For Cartan geometries admitting automorphisms with isotropies satisfying a particular, loosely dynamical property on their model geometries, we demonstrate the existence of an open subset of the geometry with trivial holonomy. This property, which generalizes characteristics of isotropies corresponding to isolated higher-order fixed points in parabolic geometries that are known to require a nearby open subset to have vanishing curvature, only relies upon the behavior of the isotropy in the model geometry, and therefore applies regardless of initial curvature assumptions, such as regularity or normality. Along the way to proving our main results, we also derive a couple of results for working with holonomy, relating to limits of sequences of developments and the existence of antidevelopments, that are useful in their own right. To showcase the effectiveness of the techniques developed, we use them to completely characterize all almost c-projective and almost quaternionic structures that admit a nontrivial automorphism with a higher-order fixed point, as well as all nondegenerate partially integrable almost CR structures that admit a higher-order fixed point with non-null isotropy.