Spectral Schwarz Lemmas and Parallel Obstructions to Metric Comparison
In this paper, we characterize when comparison of contracted Chern curvatures implies comparison of Hermitian bundle metrics with the same constant on compact Hermitian bases. If the curvature contraction of the reference metric is semipositive, this implication holds for every Hermitian metric if and only if the kernel of this contraction contains no nonzero subbundle parallel with respect to the Chern connection of the reference metric. The equality terms of a $C^2$ spectral truncation identify the parallel summand on which comparison can fail, without regularity assumptions on the curvature kernel. We also establish a sharp operator estimate, with a constant independent of the bundle ranks, for holomorphic bundle maps on complete Kähler bases under curvature bounds and bounded base-map energy density. Smooth Schatten moments give global control without a separate boundedness assumption on the bundle map. Finally, we obtain rank bounds and conditions for vanishing on compact Hermitian bases from directional curvature traces over fiber subspaces. For higher-dimensional subspaces, these conditions allow mixed signs on individual fiber lines.