Members of VBAC (Vector Bundles on Algebraic Curves), which is partially supported by EAGER (EC FP5 Contract no. HPRN-CT-2000-00099) and by EDGE (EC FP5 Contract no. HPRN-CT-2000-00101). 2 Partially supported by the National Science Foundation under grant DMS-0072073 3 Partially supported by the Ministerio de Ciencia y Tecnología (Spain) under grant BFM2000-0024 4 Partially supported by the Fundação para a Ciência e a Tecnologia (Portugal) through the Centro de Matemática da Universidade do Porto and through grant no. SFRH/BPD/1606/2000.

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

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, which we explore in a companion paper "Surface group representations in PU(p,q) and Higgs bundles". 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. This paper, and its companion mentioned above, form a substantially revised version of math.AG/0206012.Comment: 44 pages. v2: minor clarifications and corrections, to appear in Math. Annale

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| 2g − 2. This allows one to write the moduli space as a union of subspaces indexed by d, each of which is a union of connected components. The main result of this paper is that the subspaces corresponding to d = ±(2g − 2) are connected in the case of representations in SU(2, 2), while they break up into 3 · 2 2g + 2g − 4 connected components in the case of representations in Sp(4, R). We obtain our results using the interpretation of the moduli space of representations as a moduli space of Higgs bundles, and an important step is an identification of certain subspaces as moduli spaces of stable triples, as studied by Bradlow and García-Prada.

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