Holonomy of Parabolic Geometries Near Isolated Higher-Order Fixed Points

Abstract

For Cartan geometries admitting an automorphism with isotropy 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.