On very stablity of principal G-bundles

Zelaci, Hacen

doi:10.1007/s10711-019-00447-z

Cited by 4 publications

(4 citation statements)

References 10 publications

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“…In this article we extend [PaPe, Z] to the parabolic bundle setup. Using the results in both these articles, and most crucially [Z,Lemma 1.3], we prove the equality of the wobbly and the shaky locus. It next follows from the techniques developed in [PaPe] that in the smooth moduli space case, the shaky locus can in fact be characterised in terms of a resolution of r.…”

Section: Introductionmentioning

confidence: 91%

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Wobbly and shaky bundles and resolutions of rational maps

Peón-Nieto

2020

Preprint

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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.

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Section: Introductionmentioning

confidence: 91%

“…It is worth noting that finiteness follows from very stability as a corollary of [Z,Lemma 1.3]. Moreover, in the case of nilpotent Higgs fields, properness of h E,nilp is equivalent to properness of h E := h P,α | VE (cf.…”

Section: Introductionmentioning

confidence: 99%

Wobbly and shaky bundles and resolutions of rational maps

Peón-Nieto

2020

Preprint

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“…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].…”

Section: The Case Of Genus 2 and Rankmentioning

confidence: 99%

Parabolic Hecke eigensheaves

DONAGI,

PANTEV

2022

ast

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Following the strategy outlined in [DP09, DP22] for bundles of rank 2 on a smooth projective curve of genus 2, we construct flat connections over the moduli of stable bundles, with singularities along the wobbly locus. We verify that the associated Dmodules are Hecke eigensheaves. The local systems are constructed by the nonabelian Hodge correspondence from Higgs bundles. The spectral varieties of the Higgs bundles are the Hitchin fibers corresponding to the Hecke eigenvalues.

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“…More recently, criteria for wobbliness of fixed points in terms of properness of the Hitchin map [PPe1,Z,HH] has opened the way to the computation of multiplicities of the irreducible components of the nilpotent cone [HH], only known until then for the moduli space of bundles [BNR] and the Hitchin section [BR]. Indeed, downward flows to very stable fixed points are proper subvarieties of the moduli space, intersecting the nilpotent cone with generic multiplicity [HH].…”

Section: Introductionmentioning

confidence: 99%

Wobbly moduli of chains, equivariant multiplicities and $\mathrm{U}(n_0,n_1)$-Higgs bundles

Peón-Nieto¹

2023

Preprint

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We give a birational description of the reduced schemes underlying the irreducible components of the nilpotent cone and the C × -fixed point locus of length two in the moduli space of Higgs bundles. By producing criteria for wobbliness, we are able to determine wobbly fixed point components of type (n 0 , n 1 ) and prove that these are precisely U(n 0 , n 1 )-wobbly components. We compute the virtual equivariant multiplicities of fixed points as defined by Hausel-Hitchin and find that they are polynomial for all partitions other than (2, 1) and (4, 3). In particular, this proves that they provide an obstruction to the existence of very stable fixed points only for very specific components.

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On very stablity of principal G-bundles

Cited by 4 publications

References 10 publications

Wobbly and shaky bundles and resolutions of rational maps

Wobbly and shaky bundles and resolutions of rational maps

Parabolic Hecke eigensheaves

Wobbly moduli of chains, equivariant multiplicities and $\mathrm{U}(n_0,n_1)$-Higgs bundles

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