Peter B. Gothen

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We consider the moduli spaces of representations of the fundamental group of a surface of genus g > 2 in the Lie groups SU(2, 2) and Sp(4, R). It is well known that there is a characteristic number, d , of such a representation, satisfying the inequality |d| 6 2g − 2. This allows one to write the moduli space as a union of subspaces indexed by d , each of(More)
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(More)
Using the L 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(More)
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(More)
We count the connected components in the moduli space of PU(p, q)-representations of the fundamental group for a closed oriented surface. The components are labelled by pairs of integers which arise as topological invariants of the flat bundles associated to the representations. Our results show that for each allowed value of these invariants, which are(More)
Using the L 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. We prove that these moduli(More)
A good way to understand an object of study is, as Richard Feynman famously remarked, to “just look at the thing!”. In this paper we apply Feynman’s method to answer the following question: given a surface group representation in Sp(4,R), under what conditions can it be deformed to a representation which factors through a proper reductive subgroup of(More)
The category of representations of a finite quiver in the category of sheaves of modules on a ringed space is abelian. We show that this category has enough injectives by constructing an explicit injective resolution. From this resolution we deduce a long exact sequence relating the Ext groups in these two categories. We also show that under some(More)
The Betti numbers of moduli spaces of representations of a universal central extension of a surface group in the groups U(2, 1) and SU(2, 1) are calculated. The results are obtained using the identification of these moduli spaces with moduli spaces of Higgs bundles, and Morse theory, following Hitchin’s programme [14]. This requires a careful analysis of(More)