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A Hitchin–Kobayashi correspondence for Kaehler fibrations
 J. REINE ANGEW. MATH
, 1999
"... Let X be a compact Kaehler manifold and E → X a principal K bundle, where K is a compact connected Lie group. Let A 1,1 be the set of connections on E whose curvature lies in Ω 1,1 (E ×Ad k). Let k = Lie(K), and fix on k a nondegenerate biinvariant bilinear pairing. This allows to identify k ≃ k ∗ ..."
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Cited by 42 (4 self)
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Let X be a compact Kaehler manifold and E → X a principal K bundle, where K is a compact connected Lie group. Let A 1,1 be the set of connections on E whose curvature lies in Ω 1,1 (E ×Ad k). Let k = Lie(K), and fix on k a nondegenerate biinvariant bilinear pairing. This allows to identify k ≃ k ∗. Let F be a Kaehler left Kmanifold and suppose that there exists a moment map µ: F → k ∗ for the action of K on F. Let S = Γ(E ×K F). In this paper we study the equation ΛFA + µ(Φ) = c for A ∈ A 1,1 on E and a section Φ ∈ S, where FA is the curvature of A and c ∈ k is a fixed central element. We study which orbits of the action of the complex gauge group on A 1,1 ×S contain solutions of the equation and we define a positive functional on A 1,1 ×S which generalises the YangMillsHiggs functional and whose local minima coincide with the solutions of the equation.
Surface group representations and U(p, q)Higgs bundles
, 2002
"... Using the L² norm of the Higgs field as a Morse function, we study the moduli spaces of U(p, q)Higgs bundles over a Riemann surface. We require that the genus of the surface be at least two, but place no constraints on (p, q). A key step is the identification of the function’s local minima as modul ..."
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Cited by 41 (9 self)
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Using the L² norm of the Higgs field as a Morse function, we study the moduli spaces of U(p, q)Higgs bundles over a Riemann surface. We require that the genus of the surface be at least two, but place no constraints on (p, q). A key step is the identification of the function’s local minima as moduli spaces of holomorphic triples. In a companion paper [7] we prove that these moduli spaces of triples are nonempty and irreducible. Because of the relation between flat bundles and fundamental group representations, we can interpret our conclusions as results about the number of connected components in the moduli space of semisimple PU(p, q)representations. The topological invariants of the flat bundles are used to label subspaces. These invariants are bounded by a Milnor–Wood type inequality. For each allowed value of the invariants satisfying a certain coprimality condition, we prove that the corresponding subspace is nonempty and connected. If the coprimality condition does not hold, our results apply to the closure of the moduli space of irreducible representations
BETTI NUMBERS OF THE MODULI SPACE OF RANK 3 PARABOLIC HIGGS BUNDLES
, 2004
"... Abstract. Parabolic Higgs bundles on a Riemann surface are of interest for many reasons, one of them being their importance in the study of representations of the fundamental group of the punctured surface in the complex general linear group. In this paper we calculate the Betti numbers of the modul ..."
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Cited by 31 (8 self)
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Abstract. Parabolic Higgs bundles on a Riemann surface are of interest for many reasons, one of them being their importance in the study of representations of the fundamental group of the punctured surface in the complex general linear group. In this paper we calculate the Betti numbers of the moduli space of rank 3 parabolic Higgs bundles, using Morse theory. A key point is that certain critical submanifolds of the Morse function can be identified with moduli spaces of parabolic triples. These moduli spaces come in families depending on a real parameter and we carry out a careful analysis of them by studying their variation with this parameter. Thus we obtain in particular information about the topology of the moduli spaces of parabolic triples for the value of the parameter relevant to the study of parabolic Higgs bundles. The remaining critical submanifolds are also described: one of them is the moduli space of parabolic bundles, while the remaining ones have a description in terms of symmetric products of the Riemann surface. As another consequence of our Morse theoretic analysis, we obtain a proof of the parabolic version of a theorem of Laumon, which states that the nilpotent cone (the preimage of zero under the Hitchin map) is a Lagrangian subvariety of the moduli space of parabolic Higgs bundles. 1.
Homological algebra of twisted quiver bundles
 J. London Math. Soc
"... Several important cases of vector bundles with extra structure (such as Higgs bundles and triples) may be regarded as examples of twisted representations of a finite quiver in the category of sheaves of modules on a variety/manifold/ringed space. We show that the ..."
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Cited by 28 (3 self)
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Several important cases of vector bundles with extra structure (such as Higgs bundles and triples) may be regarded as examples of twisted representations of a finite quiver in the category of sheaves of modules on a variety/manifold/ringed space. We show that the
Moduli spaces of holomorphic triples over compact Riemann surfaces
, 2002
"... A holomorphic triple over a compact Riemann surface consists of two holomorphic vector bundles and a holomorphic map between them. After fixing the topological types of the bundles and a real parameter, there exist moduli spaces of stable holomorphic triples. In this paper we study nonemptiness, ir ..."
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Cited by 24 (7 self)
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A holomorphic triple over a compact Riemann surface consists of two holomorphic vector bundles and a holomorphic map between them. After fixing the topological types of the bundles and a real parameter, there exist moduli spaces of stable holomorphic triples. In this paper we study nonemptiness, irreducibility, smoothness, and birational descriptions of these moduli spaces for a certain range of the parameter. Our results have important applications to the study of the moduli space of representations of the fundamental group of the surface into unitary Lie groups of indefinite signature ([5, 7]). Another application, that we study in this paper, is to the existence of stable bundles on the product of the surface by the complex projective line.
DIMENSIONAL REDUCTION AND QUIVER BUNDLES
, 2002
"... The socalled Hitchin–Kobayashi correspondence, proved by Donaldson, Uhlenbeck and Yau, establishes that an indecomposable holomorphic vector bundle over a compact Kähler manifold admits a Hermitian–Einstein metric if and only if the bundle satisfies the Mumford–Takemoto stability condition. In this ..."
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Cited by 19 (0 self)
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The socalled Hitchin–Kobayashi correspondence, proved by Donaldson, Uhlenbeck and Yau, establishes that an indecomposable holomorphic vector bundle over a compact Kähler manifold admits a Hermitian–Einstein metric if and only if the bundle satisfies the Mumford–Takemoto stability condition. In this paper we consider a variant of this correspondence for Gequivariant vector bundles on the product of a compact Kähler manifold X by a flag manifold G/P, where G is a complex semisimple Lie group and P is a parabolic subgroup. The modification that we consider is determined by a filtration of the vector bundle which is naturally defined by the equivariance of the bundle. The study of invariant solutions to the modified Hermitian–Einstein equation over X × G/P leads, via dimensional reduction techniques, to gaugetheoretic equations on X. These are equations for hermitian metrics on a set of holomorphic bundles on X linked by morphisms, defining what we call a quiver bundle for a quiver with relations whose structure is entirely determined by the parabolic subgroup P. Similarly, the corresponding stability condition for the invariant filtration over X × G/P gives rise to a stability condition for the quiver bundle on X, and hence to a Hitchin–Kobayashi correspondence. In the simplest case, when the flag manifold is the complex projective line, one recovers the theory of vortices, stable triples and stable chains, as studied by Bradlow, the authors, and others.
HITCHIN–KOBAYASHI CORRESPONDENCE, QUIVERS, AND VORTICES
, 2003
"... A twisted quiver bundle is a set of holomorphic vector bundles over a complex manifold, labelled by the vertices of a quiver, linked by a set of morphisms twisted by a fixed collection of holomorphic vector bundles, labelled by the arrows. When the manifold is Kähler, quiver bundles admit natural ..."
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Cited by 15 (1 self)
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A twisted quiver bundle is a set of holomorphic vector bundles over a complex manifold, labelled by the vertices of a quiver, linked by a set of morphisms twisted by a fixed collection of holomorphic vector bundles, labelled by the arrows. When the manifold is Kähler, quiver bundles admit natural gaugetheoretic equations, which unify many known equations for bundles with extra structure. In this paper we prove a Hitchin–Kobayashi correspondence for twisted quiver bundles over a compact Kähler manifold, relating the existence of solutions to the gauge equations to a stability criterion, and consider its application to a number of situations related to Higgs bundles and dimensional reductions of the Hermitian–Einstein equations.