$C^{\infty}$ Stability, Canonical Maps, and Discrete Dynamics

TL;DR

This paper investigates a discrete dynamical system aimed at identifying the most holomorphic connection on complex vector bundles, exploring geometric relations, singularities, and curvature expressions in the context of canonical connections.

Contribution

It introduces a novel discrete dynamical system for finding canonical connections and analyzes their geometric properties, including curvature and singularity formation.

Findings

Relation between Chern classes and singularities

Expression of curvature via heat kernels

Role of canonical metrics from Grassmannians

Abstract

We study a discrete dynamical system designed to find a 'most holomorphic' connection on a smooth complex vector bundle $E$. We examine the relation between the distance of the chern classes of $E$ from the $(p,p)$ axis of the Hodge diamond and singularity formation. Canonical connections and canonical metrics pulled back from Grassmannians play a major role, and we review their differential geometry. As an exercise in the geometry of canonical connections, we include an expression for the curvature in terms of heat kernels.