Let X be a smooth projective curve of genus g > 1 over an algebraically closed field of positive characteristic. This paper is a study of a natural stratification, defined by the absolute Frobenius morphism of X, on the moduli space of vector bundles. In characteristic two, there is a complete classification of semi-stable bundles of rank 2 which are destabilized by Frobenius pull-back. We also show that these strata are irreducible and obtain their respective dimensions. In particular, the dimension of the locus of bundles of rank two which are destabilized by Frobenius is 3g − 4. These Frobenius destabilized bundles also exist in characteristics two, three and five with ranks 4, 3 and 5, respectively. Finally, there is a connection between (pre)-opers and Frobenius destabilized bundles. This allows an interpretation of some of the above results in terms of pre-opers and provides a mechanism for constructing Frobenius destabilized bundles in large characteristics.

For a compact Riemann surface X of genus g > 1, Hom(π 1 (X), U(p, 1))/ U(p, 1) is the moduli space of flat U(p, 1)connections on X. There is an integer invariant, τ , the Toledo invariant associated with each element in Hom(π 1 (X), U(p, 1))/ U(p, 1). If q = 1, then −2(g − 1) ≤ τ ≤ 2(g − 1). This paper shows that Hom(π 1 (X), U(p, 1))/ U(p, 1) has one connected component corresponding to each τ ∈ 2Z with −2(g − 1) ≤ τ ≤ 2(g − 1). Therefore the total number of connected components is 2(g − 1) + 1.

Abstract. We prove that the mapping class group of a closed surface acts ergodically on connected components of the representation variety corresponding to a connected compact Lie group.

This expository article details the theory of rank one Higgs bundles over a closed Riemann surface X and their relation to representations of the fundamental group of X. We construct an equivalence between the deformation theories of flat connections and Higgs pairs. This provides an identification of moduli spaces arising in different contexts. The moduli spaces are real Lie groups. From each context arises a complex structure, and the different complex structures define a hyperkähler structure. The twistor space, real forms, and various group actions are computed explicitly in terms of the Jacobian of X. We describe the moduli spaces and their geometry in terms of the Riemann period matrix of X. This is the simplest case of the theory developed by Hitchin, Simpson and others. We emphasize its formal aspects that generalize to higher rank Higgs bundles over higher dimensional Kähler manifolds. To the memory of Hsieh Po-Hsunv vi CONTENTS 7. The moduli space and the Riemann period matrix 7.1. Coordinates for the Betti moduli space 7.2. Abelian differentials and their periods 7.3. Flat connections 7.4. Higgs fields 7.5. The C * -action in terms of the period matrix 7.6. The C * -action and the real points Bibliography Rank One Higgs Bundles

Let M be a one-holed torus with boundary ∂M (a circle) and Γ the mapping class group of M fixing ∂M . The group Γ acts on M C (SU (2)) which is the space of SU(2)-gauge equivalence classes of flat SU(2)-connections on M with fixed holonomy on ∂M . We study the topological dynamics of the Γ-action and give conditions for the individual Γ-orbits to be dense in M C (SU(2)).