Abelian and non-abelian cohomology

We place the representation variety in the broader context of abelian and nonabelian cohomology. We outline the equivalent constructions of the moduli spaces of flat bundles, of smooth integrable connections, and of holomorphic integrable connections over a compact Kaehler manifold. In addition, we describe the moduli space of Higgs bundles and how it relates to the representation variety. We attempt to avoid abstraction, but strive to present and clarify the unifying ideas underlying the theory.Comment: 39 pages. Lecture Notes Series, Institute for Mathematical Sciences, National University of Singapore: Volume 23, Geometry, Topology And Dynamics Of Character Varieties, 2012 Geometry, Topology And Dynamics Of Character Varietie

The Moduli of Flat U(p,1) Structures on Riemann Surfaces

For a compact Riemann surface $X$ of genus $g > 1$, \Hom(\pi_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, $\tau$, the Toledo invariant associated with each element in \Hom(\pi_1(X), U(p,1))/U(p,1). If $q = 1$, then $-2(g-1) \le \tau \le 2(g-1)$. This paper shows that \Hom(\pi_1(X), U(p,1))/U(p,1) has one connected component corresponding to each $\tau \in 2Z$ with $-2(g-1) \le \tau \le 2(g-1)$. Therefore the total number of connected components is $2(g-1) + 1$.Comment: 12 pages. The revised version corrects a technical mistake in the previous version in section 4.

The algebraic de Rham cohomology of representation varieties

The SL(2,C)-representation varieties of punctured surfaces form natural families parameterized by holonomies at the punctures. In this paper, we first compute the loci where these varieties are singular for the cases of one-holed and two-holed tori and the four-holed sphere. We then compute the de Rham cohomologies of these varieties of the one-holed torus and the four-holed sphere when the varieties are smooth via the Grothendieck theorem. Furthermore, we produce the explicit Gauss-Manin connection on the natural family of the smooth SL(2,C)-representation variety of the one-holed torus.Comment: Minor stylistic revision from version 1, 21 page

Explicit Connections with SU(2)-Monodromy

The pure braid group \Gamma of a quadruply-punctured Riemann sphere acts on the SL(2,C)-moduli M of the representation variety of such sphere. The points in M are classified into \Gamma-orbits. We show that, in this case, the monodromy groups of many explicit solutions to the Riemann-Hilbert problem are subgroups of SU(2). Most of these solutions are examples of representations that have dense images in SU(2), but with finite \Gamma-orbits in M. These examples relate to explicit immersions of constant mean curvature surfaces.Comment: 6 pages. Corrected a few typographical errors in the previous versio

Strong Lefschetz property under reduction

Let n>1 and G be the group SU(n) or Sp(n). This paper constructs compact symplectic manifolds whose symplectic quotient under a Hamiltonian G-action does not inherit the strong Lefschetz property.Comment: 9 pages. Added some computation detail

Ergodicity of Mapping Class Group Actions on Representation Varieties, II. Surfaces with Boundary

The mapping class group of a compact oriented surface of genus greater than one with boundary acts ergodically on connected components of the representation variety corresponding to a connected compact Lie group, for every choice of conjugacy class boundary condition.Comment: 5 pages and 1 figur

Moduli of vector bundles on curves in positive characteristic

Let $X$ be a projective curve of genus 2 over an algebraically closed field of characteristic 2. The Frobenius map on X induces a rational map on the moduli space of rank-2 bundles. We show that up to isomorphism, there is only one (up to tensoring by an order two line bundle) semi-stable vector bundle of rank 2 with determinant equal to a theta characteristic whose Frobenius pull-back is not stable. The indeterminacy of the Frobenius map at this point can be resolved by introducing Higgs bundles.Comment: AmsLaTeX file (10 printed pages

Topological Dynamics on Moduli Spaces, I

Let M be a one-holed torus with boundary $\partial M$ (a circle) and $\Gamma$ the mapping class group of M fixing $\partial M$. The group $\Gamma$ acts on ${\mathcal M}_{\mathcal C}(SU(2))$ which is the space of SU(2)-gauge equivalence classes of flat SU(2)-connections on M with fixed holonomy on $\partial M$. We study the topological dynamics of the $\Gamma$-action and give conditions for the individual $\Gamma$-orbits to be dense in ${\mathcal M}_{\mathcal C}(SU(2))$.Comment: 22 pages, 1 figure in Postscript forma

Rank One Higgs Bundles and Representations of Fundamental Groups of Riemann Surfaces

This expository paper details the theory of rank one Higgs bundles over a closed Riemann surface X and their relationship 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 hyper-Kaehlerstructure. 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 Kaehler manifolds

Action of the Johnson-Torelli group on Representation Varieties

Let \Sigma be a compact orientable surface with genus g and n boundary components B = (B_1,..., B_n). Let c = (c_1,...,c_n) in [-2,2]^n. Then the mapping class group MCG of \Sigma acts on the relative SU(2)-character variety X_c := Hom_C(\pi, SU(2))/SU(2), comprising conjugacy classes of representations \rho with tr(\rho(B_i)) = c_i. This action preserves a symplectic structure on the smooth part of X_c, and the corresponding measure is finite. Suppose g = 1 and n = 2. Let J be the subgroup of MCG generated by Dehn twists along null homologous simple loops in \Sigma. Then the action of J on X_c is ergodic for almost all c.Comment: This new version includes a ribbon graph for clarity and corrects some typographic errors and a few misstatements between Remark 3.1 and Lemma 3.2 and between Remark 3.7 and Lemma 3.8 in the previous versio