arXiv subjects
Publications and source records attributed to Eugene Z. Xia.
We study the topology of the SU(2)-representation variety of the compact oriented surface of genus 2 with one boundary component about which the holonomy is a generator of the center of SU(2).
Let $Σ$ be a compact orientable surface of genus $g=1$ with $n=1$ boundary component. The mapping class group $Γ$ of $Σ$ acts on the SU(3)-character variety of $Σ$. We show that the action is ergodic with respect to the natural symplectic measure on the character variety.
We study the symplectic geometry of the SU(2)-representation variety of the compact oriented surface of genus 2. We use the Goldman flows to identify subsets of the moduli space with corresponding subsets of $\mathbb P^3(\mathbb C)$. We also define and study two antisymplectic involutions on the moduli space and their fixed point sets.
A degeneration of compact Kaehler manifolds gives rise to a monodromy action on Betti moduli space H^1(X, G) = Hom(π_1(X),G)/G over smooth fibres with a complex algebraic structure group G being either abelian or reductive. Assume that the singularities of the central fibre is of normal crossing. When G = C, the invariant cohomology classes arise from the global classes. This is no longer true in general. In this paper, we produce large families of locally invariant classes that do not arise from global ones for reductive G. These examples exist even when G is abelian, as long as G contains multiple torsion points. Finally, for general G, we make a new conjecture on local invariant classes and produce some suggestive examples.
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.
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.
Let Σbe 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 Σacts on the relative SU(2)-character variety X_c := Hom_C(π, SU(2))/SU(2), comprising conjugacy classes of representations ρwith tr(ρ(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 Σ. Then the action of J on X_c is ergodic for almost all c.
The pure braid group Γof 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 Γ-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 Γ-orbits in M. These examples relate to explicit immersions of constant mean curvature surfaces.
Let S be a compact orientable surface with genus g and n boundary components d_1,...,d_n. Let b = (b_1, ..., b_n) where b_n lies in [-2,2]. Then the mapping class group of S acts on the relative SU(2)-character variety X comprising conjugacy classes of representations f of the fundamental group F of S, where trace f(d_i) = b_i. This action preserves a symplectic structure on the open dense smooth submanifold of X. corresponding to irreducible representations. This subset has full measure and is connected. In this note we use the symplectic geometry of this space to give a new proof that this action is ergodic.
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.
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.
Let f be a degeneration of Kahler manifolds. The local invariant cycle theorem states that for a smooth fiber of the degeneration, any cohomology class, invariant under the monodromy action, rises from a global cohomology class. Instead of the classical cohomology, one may consider the non-abelian cohomology. This note demonstrates that the analogous non-abelian version of the local invariant cycle theorem does not hold if the first non-abelian cohomology is the moduli space (universal categorical quotient) of the representations of the fundamental group.
Let $M$ be a four-holed sphere and $Γ$ the mapping class group of $M$ fixing the boundary $\partial M$. The group $Γ$ acts on $M_B(SL(2,C)) = Hom_B^+(pi_1(M),SL(2,C))/SL(2,C)$ which is the space of completely reducible $SL(2,C)$-gauge equivalence classes of flat $SL(2,C)$-connections on $M$ with fixed holonomy $B$ on $\partial M$. Let $B \in (-2,2)^4$ and $M_B$ be the compact component of the real points of $M_B(SL(2,C))$. These points correspond to SU(2)-representations or $SL(2,R)$-representations. The $Γ$-action preserves $M_B$ and we study the topological dynamics of the $Γ$-action on $M_B$ and show that for a dense set of holonomy $B \in (-2,2)^4$, the $Γ$-orbits are dense in $M_B$. We also produce a class of representations $ρ\in \Hom_B^+(pi_1(M),SL(2,R))$ such that the $Γ$-orbit of $[ρ]$ is finite in the compact component of $M_B(SL(2,R))$, but $ρ(π_1(M))$ is dense in $SL(2,R)$.
Let X be an irreducible smooth projective curve of genus at least two over an algebraically closed field k of characteristic p>0. In this paper we study the natural stratification, defined using the absolute Frobenius of X, on the moduli space of vector bundles on X of suitable rank. In characteristic two we provide a complete classification of rank two semi-stable vector bundles whose Frobenius pull-back is not semi-stable. We also obtain fairly good information about the strata of the Frobenius stratification, including the irreducibility and the dimension of each non-empty Frobenius stratum. In particular we show that the locus of Frobenius destabilized bundles has dimension 3g-4 in the moduli space of semi-stable bundles of rank two. We also construct stable bundles that are destabilized by Frobenius in the following situations: characteristic p=2 and rank four, (2) characteristic p=rank=3, (3) characteristic p=rank=5 and g at least three. We also explore (in any characteristic) the connection between Frobenius destabilized bundles and (pre)-opers, this approach allows us to reinterpret some of our results in terms of pre-opers and also allows us to construct Frobenius destablised bundles from certain pre-opers (or opers). The other result we obtain is (for characteristic two): we show that the Gunning bundle descends under Frobenius when genus g is even. If g is odd, then the Gunning bundle twisted by any odd degree line bundle also descends.
Let X be a smooth projective curve of genus g>1 over an algebraically closed field of characteristic 2. Pull-back by the (absolute) Frobenius on X only defines a rational morphism on the moduli scheme of rank-2 vector bundles on X, because the Frobenius pull-back may destory stability of a vector bundle. This paper introduces and studies a Harder-Narasimhan type stratification on the moduli scheme and proves that the family of semi-stable rank-2 vector bundles (with a fixed degree) whose Frobenius pull-back are not semi-stable is parameterized by an irreducible subscheme of dimension 3g-4.
For a compact Riemann surface $X$ of genus $g > 1$, $\Hom(π_1(X), PU(p,q))/PU(p,q)$ is the moduli space of flat $PU(p,q)$-connections on $X$. There are two invariants, the Chern class $c$ and the Toledo invariant $τ$ associated with each element in the moduli. The Toledo invariant is bounded in the range $-2min(p,q)(g-1) \le τ\le 2min(p,q)(g-1)$. This paper shows that the component, associated with a fixed $τ> 2(max(p,q)-1)(g-1)$ (resp. $τ< -2(max(p,q)-1)(g-1)$) and a fixed Chern class $c$, is connected (The restriction on $τ$ implies $p=q$).
Let $M$ be a four-holed sphere and $Γ$ the mapping class group of $M$ fixing $\partial M$. The group $Γ$ acts on the space ${\mathcal M}_{\mathcal B}(SU(2))$ of SU(2)-gauge equivalence classes of flat SU(2)-connections on $M$ with fixed holonomy on $\partial M$. We give examples of flat SU(2)-connections whose holonomy groups are dense in SU(2), but whose $Γ$-orbits are discrete in ${\mathcal M}_{\mathcal B}(SU(2))$. This phenomenon does not occur for surfaces with genus greater than zero.
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.