Equality of the Wobbly and Shaky Loci

Let X be a smooth complex projective curve of genus g≥ 2, and let D⊂ X be a reduced divisor. We prove that a parabolic vector bundle E on X is (strongly) wobbly, that is, E has a non-zero (strongly) parabolic nilpotent Higgs field, if and only if it is (strongly) shaky, that is, 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 vector bundle moduli space, both assumed to be smooth. This solves a conjecture by Donagi–Pantev [ 14] 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.