Eugene Z. Xia

On vector bundles destabilized by Frobenius pull-back

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.

The moduli of flat PU(2,1) structures on Riemann surfaces

For X a smooth projective curve over ℂ of genus g > 1, Hom+(π1(X), U(p, 1))/U(p, 1) is the moduli space of flat semi-simple U(p, 1)-connections on X. There is an integer invariant, τ, the Toledo invariant associated with each element in Hom+(π1(X), U(p, 1))/U(p, 1). This paper shows that Hom+(π1(X), U(p, 1))/U(p, 1) has one connected component corresponding to each τ & in 2 ℤ with −2(g−1) ≤τ≤2(g−1). Therefore the total number of connected components is 2(g−1)+1.

Topological dynamics on moduli spaces II

Let M be an orientable genus g > 0 surface with boundary ∂M. Let Γ be the mapping class group of M fixing ∂M. The group Γ acts on M c = Homc(π 1 (M),SU(2))/SU(2), 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 .

Moduli of Vector Bundles on Curves in Positive Characteristics

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 scheme 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 semi-stable. The indeterminacy of the Frobenius map at this point can be resolved by introducing Higgs bundles.

Action of the Johnson-Torelli group on representation varieties

Let Σ be a compact orientable surface with genus g and n boundary components B = (B1, . . . , Bn). Let c = (c1, . . . , cn) ∈ [−2, 2]n. Then the mapping class group MCG of Σ acts on the relative SU(2)-character variety XC := HomC(π,SU(2))/SU(2), comprising conjugacy classes of representations ρ with tr(ρ(Bi)) = ci. This action preserves a symplectic structure on the smooth part of XC , and the corresponding measure is finite. Suppose g = 1 and n = 2. Let J ⊂ MCG be the subgroup generated by Dehn twists along null homologous simple loops in Σ. Then the action of J on XC is ergodic for almost all c.

Non-abelian local invariant cycles

Let f be a degeneration of Kahler manifolds. The lo- cal invariant cycle theorem states that for a smooth fiber of the de- generation, 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. A degeneration of Kahler manifolds is a proper map f from a Kahler manifold X onto the unit disksuch that f is of maximum rank for all s ∈ � except at the point s = 0. Let � � = � − {0}. We call Xt = f 1 (Xt) a smooth fiber or generic fiber when t ∈ � � and X0 = f 1 (0) the singular or degenerated fiber. We assume the singularity in X0 is of normal crossing.

The \({\textit{SU}}(2)\)-character variety of the closed surface of genus 2

We study the symplectic geometry of the

Explicit connections with SU(2)-monodromy

{\text {SU}}(2)