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

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