2026/07/21 by Mahmud Azam, Steven Rayan
#math.DG #math.AG #math.AP #math.CT #math.RT
Let X be a compact Kähler manifold. In prior work, we constructed diffeological moduli stacks of Higgs and flat bundles on X, related by extension completion of smooth harmonic families. Here, we develop the relative analytic theory. On Sobolev completions over arbitrary plots, we prove that every smooth stable Higgs family satisfying the numerical conditions admits a global smooth harmonic metric. Fixing a Hermitian--Einstein determinant metric removes scalar freedom, and then elliptic regularity and normalized gluing yield plotwise smoothness. The theorem holds at every finite parameter regularity Cd and on reduced singular parameter spaces with ambient extensions. For a Higgs deformation η, the normalized metric variation satisfies Lhs=-\mathcal Sh(η) and s=-Gh\mathcal Sh(η) up to an independent rank-one determinant term for GLr. This computes the plotwise differential and recovers the classical comparison. Locally split, constant-type polystable families admit smooth harmonic metrics. Real-analytic examples show general polystable families may have neither continuous harmonic metrics nor relative harmonic filtrations and may lie outside every Cd extension-generated locus. In one example a singular harmonic reduction produces a continuous adjoint Higgs field and a flat family with semisimple slices. This defines a weak C0 operator-level harmonic mediator, strictly larger than the metric-regular one, whose endpoint images after finite extension completion and stackification satisfy \mathscr MDol,0wk\mathcal H(X)≃\mathscr MdR,0wk\mathcal H(X). We characterize the extension-generated stack by relative harmonic filtrations, develop their obstruction theory, analyze the loss of extension data under heat flow, and construct the smooth Hodge λ-family on the stable locus.