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    ABSTRACT: Higgs bundles over a closed orientable surface can be defined for any real reductive Lie group G. In this paper we examine the case G=SO*(2n). We describe a rigidity phenomenon encountered in the case of maximal Toledo invariant. Using this and Morse theory in the moduli space of Higgs bundles, we show that the moduli space is connected in this maximal Toledo case. The Morse theory also allows us to show connectedness when the Toledo invariant is zero. The correspondence between Higgs bundles and surface group representations thus allows us to count the connected components with zero and maximal Toledo invariant in the moduli space of representations of the fundamental group of the surface in SO*(2n).
    03/2013;
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    Peter B. Gothen
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    ABSTRACT: These are the lecture notes from my course in the January 2011 School on Moduli Spaces at the Newton Institute. I give an introduction to Higgs bundles and their application to the study of character varieties for surface group representations.
    09/2012;
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    PETER B. GOTHEN
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    ABSTRACT: of genus g 2. This space was introduced by Hitchin in (10) and we generalise his calculation in the rank 2 case. We consider Higgs bundles with rank and degree coprime; this condition ensures that the moduli space is smooth. We shall do the calculation for the case of bundles with fixed determinant but we also state the result for bundles with any determinant. From a principal bundle point of view the natural space to consider is the moduli space of PSL(3,C) Higgs bundles. However, as noted in (10, §5), this space has singularities and the method of calculation does not apply directly. Hitchin analyses many aspects of the geometry of M in (10), and among other things he shows that M is a hyperkahler manifold, i.e. it has complex structures I, J, and K which satisfy the identities of the quaternions. The complex structure I arises from the inter- pretation of M as the moduli space of Higgs bundles. The complex stucture J arises from an alternative description of M, which follows from results of Donaldson (7) and, more generally, Corlette (5) (see also (10)). There is a universal central extension of 1,
    International Journal of Mathematics 01/2012; 05(06). · 0.56 Impact Factor
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    Peter B. Gothen, André Oliveira
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    ABSTRACT: Let $X$ be a compact Riemann surface. A quadratic pair on $X$ consists of a holomorphic vector bundle with a quadratic form which takes values in fixed line bundle. We show that the moduli spaces of quadratic pairs of rank 2 are connected under some constraints on their topological invariants. As an application of our results we determine the connected components of the $\mathrm{SO}_0(2,3)$-character variety of $X$.
    Geometriae Dedicata 06/2011; · 0.47 Impact Factor
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    Peter B. Gothen
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    ABSTRACT: We give an overview of the work of Corlette, Donaldson, Hitchin and Simpson leading to the non-abelian Hodge theory correspondence between representations of the fundamental group of a surface and the moduli space of Higgs bundles. We then explain how this can be generalized to a correspondence between character varieties for representations of surface groups in real Lie groups G and the moduli space of G-Higgs bundles. Finally we survey recent joint work with Bradlow, Garc\'ia-Prada and Mundet i Riera on the moduli space of maximal Sp(2n,R)-Higgs bundles.
    02/2011;
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    Peter B. Gothen, André Oliveira
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    ABSTRACT: Given any line bundle L of positive degree, on a compact Riemann surface, let $M_L^\Lambda$ be the moduli space of L-twisted Higgs pairs of rank 2 with fixed determinant isomorphic to $\Lambda$ and traceless Higgs field. We give a description of the singular fibre of the Hitchin map $H:M^L_\Lambda\to H^0(L^2)$, when the corresponding spectral curve has any singularity of type $A_{m-1}$. In particular, we prove directly that this fibre is connected.
    International Mathematics Research Notices 12/2010; 2013(5):1079-1121. · 1.12 Impact Factor
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    ABSTRACT: The main result of this paper states that there exists a residual subset of the set of critical economies whose associated equilibria are finite in number. We also show that this subset does not contain any open set and therefore the result is the best possible for our choice of topology (compact-open topology). The proof rests on results and concepts from singularity theory.
    Mathematical Social Sciences 07/2010; · 0.45 Impact Factor
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    Indranil Biswas, Peter B. Gothen, Marina Logares
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    ABSTRACT: Let $X$ be a compact Riemann surface $X$ of genus at--least two. Fix a holomorphic line bundle $L$ over $X$. Let $\mathcal M$ be the moduli space of Hitchin pairs $(E ,\phi\in H^0(End(E)\otimes L))$ over $X$ of rank $r$ and fixed determinant of degree $d$. We prove that, for some numerical conditions, $\mathcal M$ is irreducible, and that the isomorphism class of the variety $\mathcal M$ uniquely determines the isomorphism class of the Riemann surface $X$.
    Mathematical Proceedings of the Cambridge Philosophical Society 12/2009; 151. · 0.68 Impact Factor
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    Sofia B.S.D. Castro, Sami Dakhlia, Peter B. Gothen
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    ABSTRACT: We complement the Sonnenschein–Mantel–Debreu results by establishing that an exchange economy, i.e., preferences and endowments, that generates a given aggregate excess demand (AED) function is close to the economy that generates a perturbation of this AED. As a consequence, genericity and determinacy results obtained by direct perturbation of an AED are as strong as results obtained by perturbing preferences and endowments.
    Journal of Mathematical Economics. 11/2009;
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    ABSTRACT: We develop a complete Hitchin-Kobayashi correspondence for twisted pairs on a compact Riemann surface X. The main novelty lies in a careful study of the the notion of polystability for pairs, required for having a bijective correspondence between solutions to the Hermite-Einstein equations, on one hand, and polystable pairs, on the other. Our results allow us to establish rigorously the homemomorphism between the moduli space of polystable G-Higgs bundles on X and the character variety for representations of the fundamental group of X in G. We also study in detail several interesting examples of the correspondence for particular groups and show how to significantly simplify the general stability condition in these cases.
    09/2009;
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    ABSTRACT: We use Higgs bundles to answer the following question: When can a maximal Sp(4,R)-representation of a surface group be deformed to a representation which factors through a proper reductive subgroup of Sp(4,R)?
    03/2009;
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    ABSTRACT: In this paper we study the moduli space of representations of a surface group (i.e., the fundamental group of a closed oriented surface) in the real symplectic group Sp(2n,R). The moduli space is partitioned by an integer invariant, called the Toledo invariant. This invariant is bounded by a Milnor-Wood type inequality. Our main result is a count of the number of connected components of the moduli space of maximal representations, i.e. representations with maximal Toledo invariant. Our approach uses the non-abelian Hodge theory correspondence proved in a companion paper arXiv:0909.4487 [math.DG] to identify the space of representations with the moduli space of polystable Sp(2n,R)-Higgs bundles. A key step is provided by the discovery of new discrete invariants of maximal representations. These new invariants arise from an identification, in the maximal case, of the moduli space of Sp(2n,R)-Higgs bundles with a moduli space of twisted Higgs bundles for the group GL(n,R).
    09/2008;
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    ABSTRACT: We establish that an exchange economy, i.e., preferences and endowments, that generates a giiven aggregate excess demand (AED) function is close to the economy generating the AED obtained by an arbitrary perturbation of the original one.
    08/2008;
  • Notices of the American Mathematical Society 01/2007; 54(8).
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    ABSTRACT: Higgs bundles and non-abelian Hodge theory provide holomorphic methods with which to study the moduli spaces of surface group representations in a reductive Lie group G. In this paper we survey the case in which G is the isometry group of a classical Hermitian symmetric space of non-compact type. Using Morse theory on the moduli spaces of Higgs bundles, we compute the number of connected components of the moduli space of representations with maximal Toledo invariant
    Geometriae Dedicata 11/2005; 122(1):185-213. · 0.47 Impact Factor
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    ABSTRACT: We calculate certain homotopy groups of the moduli spaces for representations of a compact oriented surface in the Lie groups and . Our approach relies on the interpretation of these representations in terms of Higgs bundles and uses Bott–Morse theory on the corresponding moduli spaces.
    Topology. 07/2005;
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    O. García-Prada, P. B. Gothen, V. Muñoz
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    ABSTRACT: We compute 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 study their variation with this parameter.
    Memoirs of the American Mathematical Society 12/2004; · 1.82 Impact Factor
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    ABSTRACT: We determine the number of connected components of the moduli space for representations of a surface group in the general linear group.
    02/2004;
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    ABSTRACT: 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 non-emptiness, 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.
    Mathematische Annalen 12/2003; 328(1):299-351. · 1.38 Impact Factor
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    ABSTRACT: Using the L<sup>2</sup> 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.
    Journal of differential geometry 01/2003; · 1.18 Impact Factor

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