Equality of the wobbly and shaky loci

TL;DR

This paper proves the equivalence between wobbly and shaky parabolic vector bundles on smooth complex projective curves, confirming a conjecture and establishing criteria for stability in the context of Higgs bundles.

Contribution

It establishes the equivalence of wobbly and shaky parabolic bundles, solving a conjecture by Donagi-Pantev, and proves stability criteria for strongly very stable parabolic bundles.

Findings

Wobbly and shaky parabolic bundles are equivalent.

Proved stability of strongly very stable parabolic bundles.

Provided criteria for very stability of parabolic bundles.

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.