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

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.