Equality of the wobbly and shaky loci

Abstract: Let $X$ be a smooth complex projective curve of genus $g\geq 2$. We prove that a parabolic vector bundle $\mathcal{E}$ on $X$ on $X$ is (strongly) wobbly, i.e. $\mathcal{E}$ has a non-zero (strongly) parabolic nilpotent Higgs field, if and only if it is (strongly) shaky, i.e., it is in the image of the exceptional divisor of a suitable resolution of the rational map from the (strongly) parabolic Higgs moduli to the parabolic bundle moduli space, both assumed to be smooth. This solves a conjecture by Donagi-Pantev [DP1] in the parabolic and the vector bundle context. To this end, we prove the stability of strongly very stable parabolic bundles, and criteria for very stability of parabolic bundles.