Restricting Higgs bundles to curves

Bruzzo, Ugo; Giudice, Alessio Lo

doi:10.4310/ajm.2016.v20.n3.a1

Cited by 15 publications

(18 citation statements)

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“…Note that for a suitable choice of polarization on X 2 , the vector bundle E becomes stable (see [DDK,Proposition 6.1.5]). If discriminant of E was zero then by [BG,Theorem 2.5], we can conclude that (E ⊗O(D))| D ′ 1 is semistable, which is not the case. So discriminant of E is non-zero.…”

Section: Seshadri Constants Of Equivariant Vector Bundles On Hirzebru...mentioning

confidence: 98%

Seshadri constants of equivariant vector bundles on toric varieties

Dasgupta¹,

Khan²,

Subramaniam³

2021

Preprint

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We compute Seshadri constants of a torus equivariant nef vector bundle on a projective space satisfying certain conditions. As an application, we compute Seshadri constants of tangent bundles on projective spaces. We also consider equivariant nef vector bundles on Bott towers of height 2 (i.e. Hirzebruch surfaces) and Bott towers of height 3 respectively. Assuming some conditions on the minimal slope of the restrictions of these bundles to invariant curves, we give precise values of Seshadri constant at an arbitrary point. We also give several examples illustrating our results.

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Section: Seshadri Constants Of Equivariant Vector Bundles On Hirzebru...mentioning

confidence: 98%

Seshadri constants of equivariant vector bundles on toric varieties

Dasgupta¹,

Khan²,

Subramaniam³

2021

Preprint

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“…We will show that Proposition 2.3 can be generalized to the case of rank 2 Higgs sheaves on genus one fibered surfaces. In the proof we shall need a slight generalization of a result from [6], stating that for torsion-free Hitchin pairs, with values in a slope-semistable locally free sheaf of non-positive degree, slope-semistability as an ordinary sheaf and as a pair are, in fact, equivalent; we include a proof of the result (Proposition 3.1), for the sake of completeness and for the reader's convenience. Let us recall, first of all, the notions of Hitchin pair, and of their slope-stability.…”

Section: Rank 2 Higgs Sheaves On Genus One Fibered Surfacesmentioning

confidence: 99%

“…We proved that if the spectral curve 1 of V is reduced, then the Higgs field φ is vertical, while if the bundle V is fiberwise regular with reduced (resp., integral) spectral curve, and if its rank and second Chern number satisfy an inequality involving the genus of B and the degree of the fundamental line bundle of π (resp., if the fundamental line bundle is sufficiently ample), then φ is scalar. These results were applied to the problem of characterizing slope-semistable Higgs bundles with vanishing discriminant on the class of elliptic surfaces considered, in terms of the semistability of their pull-backs via maps from arbitrary (smooth, irreducible, complete) curves to X; i.e., we partly established, for the class of elliptic surfaces considered, the conjecture about Higgs bundles satisfying the last mentioned condition that was stated in [3] and was studied in [4,3,5,6,2].…”

Section: Introductionmentioning

confidence: 99%

Moduli of rank 2 Higgs sheaves on elliptic surfaces

Bruzzo¹,

Peragine²

2021

Preprint

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We study torsion-free, rank 2 Higgs sheaves on genus one fibered surfaces, (semi)stable with respect to suitable polarizations in the sense of Friedman and O'Grady. We prove that slope-semistability of a Higgs sheaf on the surface implies semistability on the generic fiber. In the case of Higgs sheaves of odd fiber degree on elliptic surfaces in characteristic = 2, we prove that any moduli space of Higgs sheaves with fixed numerical invariants splits canonically as the product of the moduli space of ordinary sheaves (with the same invariants), and the space of global regular 1-forms on the surface. For elliptic surfaces with section in characteristic zero, and in the case arbitrary fiber degree, we prove that if a Higgs sheaf has reduced Friedman spectral curve, or is regular on a general fiber with non-reduced spectral cover, then its Higgs field takes values in the saturation of the pull-back of the canonical bundle of the base curve in the cotangent bundle of the surface.

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“…In [8] a characterization was given of some classes of varieties for which the conjecture holds (basically, varieties with nef tangent bundle).…”

Section: Categories Of Numerically Flat Bundlesmentioning

confidence: 99%

Higgs bundles and fundamental group schemes

Biswas

Bruzzo

Gurjar

2019

Advances in Geometry

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Relying on a notion of "numerical effectiveness" for Higgs bundles, we show that the category of "numerically flat" Higgs vector bundles on a smooth projective variety X is a Tannakian category. We introduce the associated group scheme, that we call the "Higgs fundamental group scheme of X," and show that its properties are related to a conjecture about the vanishing of the Chern classes of numerically flat Higgs vector bundles.

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Restricting Higgs bundles to curves

Cited by 15 publications

References 0 publications

Seshadri constants of equivariant vector bundles on toric varieties

Seshadri constants of equivariant vector bundles on toric varieties

Moduli of rank 2 Higgs sheaves on elliptic surfaces

Higgs bundles and fundamental group schemes

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