Wobbly and shaky bundles and resolutions of rational maps

Abstract

Let X be a smooth complex projective curve of genus g ≥ 2. In this article, we prove that a stable tame parabolic vector bundle E on X is very stable, i.e. E has no non-zero nilpotent Higgs field preserving the parabolic structure, if and only if the restriction of the Hitchin map to the space of parabolic Higgs fields with nilpotent residue is a proper map. The same result holds in the setup of strongly parabolic Higgs bundles. Both results follow from [Z, Lemma 1.3], once the image of the Hitchin map restricted to suitable leaves has been proven to be affine of the right dimension. This characterisation of wobbliness proves the equivalence between this notion and shakiness. We use this and the techniques in [PaPe] to prove that shaky bundles are the image of the exceptional divisor of a suitable resolution of a certain rational map when the moduli space is smooth. As corollaries to the aforementioned results, we obtain equivalent ones for the moduli space of vector bundles.

Parabolic Hecke eigensheaves

“…The other components usually correspond to the components of the wobbly locus Wob [PP21, PN20,HH22], in the sense that the other components of h −1 (0) intersect X along components of Wob. Our case of Higgs 0 is an exception: there, Wob 0 has an extra component that does not correpond to an additional component of the nilpotent cone.…”

“…Several authors have recently investigated more closely the structure of the structure of the nilpotent cone and the wobbly locus in general situations: Bozec [Boz22], Gothen-Zúñiga-Rojas [GnR22], Pal-Pauly [PP21], Peón-Nieto [PN20,FGOPN23], Zelaci [Zel20], Hellmann [Hel21], Hausel-Hitchin [HH22].…”