A GIT interpretation of the Harder–Narasimhan filtration

Gómez, Tomás L.; Sols, Ignacio; Zamora, Alfonso

doi:10.1007/s13163-014-0149-3

Cited by 24 publications

(21 citation statements)

References 13 publications

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“…The vector which maximizes such functions verifies some properties that will be strongly related to the properties of the Harder-Narasimhan filtration. In this section we recall the results of [GSZ,Section 2].…”

Section: Results On Convexitymentioning

confidence: 99%

On the Harder-Narasimhan filtration for finite dimensional representations of quivers

Zamora

2013

Geom Dedicata

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Abstract. We prove that the Harder-Narasimhan filtration for an unstable finite dimensional representation of a finite quiver coincides with the filtration associated to the 1-parameter subgroup of Kempf, which gives maximal unstability in the sense of Geometric Invariant Theory for the corresponding point in the parameter space where these objects are parametrized in the construction of the moduli space.

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Section: Results On Convexitymentioning

confidence: 99%

On the Harder-Narasimhan filtration for finite dimensional representations of quivers

Zamora

2013

Geom Dedicata

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“…Lemma 6.5 (Lemma 3.5 of [3] or Lemma 2.1.16 of [15]). (1), [x] + = max{0, x}, and μ max (E) (respectively μ min (E)) is the maximum (resp.…”

Section: Proposition 63mentioning

confidence: 99%

“…[15]) and it is a continuation of [3]. In a moduli problem, usually, we impose a notion of stability for the objects in order to obtain a moduli space with good properties.…”

Section: Introductionmentioning

confidence: 99%

“…In [3], the authors develop an idea to answer positively the previous question and establish a correspondence between both filtrations, based on rewriting a function (which appears in [10] and is maximized by the special 1-parameter subgroup) in a more geometrical way, to show that the Kempf filtration satisfies certain convexity properties (c.f. § 5), similar to the properties which characterize the Harder-Narasimhan filtration.…”

Section: Introductionmentioning

confidence: 99%

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Harder–Narasimhan filtration for rank 2 tensors and stable coverings

Zamora

2016

Proc Math Sci

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We construct a Harder-Narasimhan filtration for rank 2 tensors, where there does not exist any such notion a priori, as coming from a GIT notion of maximal unstability. The filtration associated to the 1-parameter subgroup of Kempf giving the maximal way to destabilize, in the GIT sense, a point in the parameter space of the construction of the moduli space of rank 2 tensors over a smooth projective complex variety, does not depend on a certain integer used in the construction of the moduli space, for large values of the integer. Hence, this filtration is unique and we define the Harder-Narasimhan filtration for rank 2 tensors as this unique filtration coming from GIT. Symmetric rank 2 tensors over smooth projective complex curves define curve coverings lying on a ruled surface, hence we can translate the stability condition to define stable coverings and characterize the Harder-Narasimhan filtration in terms of intersection theory.

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“…Let (X, O X (1)) be a nonsingular polarized variety. In [6], the authors proved that for a framed sheaf (E, α : E → O X ), with E locally free, its Harder-Narasimhan filtration coincides with its Kempf filtration coming from GIT theory.…”

Section: Harder-narasimhan Filtrationmentioning

confidence: 99%