Moduli for decorated tuples of sheaves and representation spaces for quivers

Schmitt, Alexander

doi:10.1007/bf02829837

Cited by 48 publications

(67 citation statements)

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“…2.9.2.42] proves that the stability condition (2.2) is in fact the asymptotic form of a more general GIT stability condition for ADHM sheaves. Analogous stability conditions for quiver sheaves were previously formulated and studied in [73,30,74], and in the context of HitchinKobayashi correspondence in [2,1]. This is a natural generalization of previous work on decorated sheaves, including [13,79,14,15,9,41,40,75].…”

Section: Remark 12 (I)mentioning

confidence: 56%

Moduli of ADHM sheaves and the local Donaldson–Thomas theory

Diaconescu

2012

Journal of Geometry and Physics

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Abstract. The ADHM construction establishes a one-to-one correspondence between framed torsion free sheaves on the projective plane and stable framed representations of a quiver with relations in the category of complex vector spaces. This paper studies the geometry of moduli spaces of representations of the same quiver with relations in the abelian category of coherent sheaves on a smooth complex projective curve X. In particular it is proven that this moduli space is virtually smooth and related by relative Beilinson spectral sequence to the curve counting construction via stable pairs of Pandharipande and Thomas. This yields a new conjectural construction for the local Donaldson-Thomas theory of curves as well as a natural higher rank generalization.

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Section: Remark 12 (I)mentioning

confidence: 56%

Moduli of ADHM sheaves and the local Donaldson–Thomas theory

Diaconescu

2012

Journal of Geometry and Physics

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“…These higher dimensional versions have, in turn, important applications in the problem of compactifying moduli spaces of principal bundles with singular objects [40], [17]. Finally, there is now also a version for product groups GL(r 1 ) × · · · × GL(r s ) over base manifolds of arbitrary dimension [41] the construction of which is based on the results of the present paper. cepted as his habilitation thesis by the University of Essen.…”

Section: Introductionmentioning

confidence: 99%

A Universal Construction for Moduli Spaces of Decorated Vector Bundles over Curves

Schmitt

2004

Transformation Groups

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111

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Let X be a smooth projective curve over the field of complex numbers, and fix a homogeneous representation ρ : GL(r) → GL(V ). Then one can associate to every vector bundle E of rank r over X a vector bundle Eρ with fibre V . We would like to study triples (E, L, ϕ) where E is a vector bundle of rank r over X, L is a line bundle over X, and ϕ : Eρ → L is a nontrivial homomorphism. This setup comprises well known objects such as framed vector bundles, Higgs bundles, and conic bundles. In this paper, we will formulate a general (parameter dependent) semistability concept for such triples, which generalizes the classical Hilbert-Mumford criterion, and we establish the existence of moduli spaces for the semistable objects.In the examples which have been studied so far, our semistability concept reproduces the known ones. Therefore, our results give in particular a unified construction for many moduli spaces considered in the literature.

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“…Note that in other cases our notions of Gieseker κ-stability and κ-semistability seem to be different to the ones introduced by Schmitt [46], as we already mentioned. We should also remark that the notions in Definition 6 can be obtained, via dimensional reduction, from equivariant versions of Gieseker semistability and Gieseker stability for equivariant sheaves on certain varieties with a Lie group action, using the methods of [2].…”

Section: Definition 6 a Q-sheaf E On X Is Gieseker κ-Semistable If Itmentioning

confidence: 88%

“…Moduli of Higgs bundles were also constructed with methods of gauge theory (Hitchin [33]) and algebraic geometry (Nitsure [44], Simpson [49]). There are also algebrogeometric constructions for holomorphic chains (Schmitt [45]) of arbitrary length and even of quiver sheaves for arbitrary quivers (Schmitt [46]). We should remark that the stability notion used by Schmitt [46] does not coincide with the one we introduce below, with the exception of quiver bundles over projective curves, and indeed there may exist a more general stability notion which includes both as particular cases.…”

Section: Construction Of the Moduli Spacementioning

confidence: 99%

“…There are also algebrogeometric constructions for holomorphic chains (Schmitt [45]) of arbitrary length and even of quiver sheaves for arbitrary quivers (Schmitt [46]). We should remark that the stability notion used by Schmitt [46] does not coincide with the one we introduce below, with the exception of quiver bundles over projective curves, and indeed there may exist a more general stability notion which includes both as particular cases. Our notion comes directly from Gieseker stability for sheaves, with the Hilbert polynomial replaced by a linear combination of the Hilbert polynomials of the sheaves attached to the vertices (see (13) and Lemma 7).…”

Section: Construction Of the Moduli Spacementioning

confidence: 99%

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Some results on the moduli spaces of quiver bundles

Álvarez-Cónsul

2008

Geom Dedicata

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This article is concerned with the study of gauge theory, stability and moduli for twisted quiver bundles in algebraic geometry. We review natural vortex equations for twisted quiver bundles and their link with a stability condition. Then we provide a brief overview of their relevance to other geometric problems and explain how quiver bundles can be viewed as sheaves of modules over a sheaf of associative algebras and why this view point is useful, e.g., in their deformation theory. Next we explain the main steps of an algebro-geometric construction of their moduli spaces. Finally, we focus on the special case of holomorphic chains over Riemann surfaces, providing some basic links with quiver representation theory. Combined with the analysis of the homological algebra of quiver sheaves and modules, these links provide a criterion for smoothness of the moduli spaces and tools to study their variation with respect to stability.Keywords Quiver bundle · Vortex equations · Moduli space · Stability · Hitchin-Kobayashi correspondence Mathematics Subject Classification (2000) 14D20 · 16G20 · 14F05 · 32L05 · 53C07 This article is a review of several results about quiver bundles in algebraic and complex geometry, with emphasis on their gauge-theoretic equations, the algebro-geometric construction of their moduli spaces and some applications.A quiver, or directed graph, is a pair of sets Q = (Q 0 , Q 1 ) together with two maps h, t : Q 1 → Q 0 . The elements of Q 0 and Q 1 are respectively called the vertices and the arrows of the quiver. For each arrow a ∈ Q 1 , the vertices v = ta and w = ha are respectively called the tail and the head of the arrow a, which is denoted by a : v → w. We will always assume that the quiver is finite, i.e. the sets Q 0 and Q 1 are finite.Quivers provide concrete descriptions for many interesting categories in terms of their representations. A representation (E, φ) of Q in a category A is the data given by an L. Álvarez-Cónsul (B) Instituto de Ciencias Matemáticas (CSIC-UAM-UC3M-UCM), Serrano 113 bis,

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Moduli for decorated tuples of sheaves and representation spaces for quivers

Abstract: We extend the scope of a former paper to vector bundle problems involving more than one vector bundle. As the main application, we obtain the solution of the well-known moduli problems of vector bundles associated with general quivers.Subject Classification: 14D20, 16G20.

Cited by 48 publications

References 22 publications

Moduli of ADHM sheaves and the local Donaldson–Thomas theory

Moduli of ADHM sheaves and the local Donaldson–Thomas theory

A Universal Construction for Moduli Spaces of Decorated Vector Bundles over Curves

Some results on the moduli spaces of quiver bundles

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