Marina Logares

Moduli of parabolic Higgs bundles and Atiyah algebroids

Abstract In this paper we study the geometry of the moduli space of (non-strongly) parabolic Higgs bundles over a Riemann surface with marked points. We show that this space possesses a Poisson structure, extending the one on the dual of an Atiyah algebroid over the moduli space of parabolic vector bundles. By considering the case of full flags, we get a Grothendieck–Springer resolution for all other flag types, in particular for the moduli spaces of twisted Higgs bundles, as studied by Markman and Bottacin and used in the recent work of Laumon–Ngô. We discuss the Hitchin system, and demonstrate that all these moduli spaces are integrable systems in the Poisson sense.

On moduli spaces of Hitchin pairs

Let X be a compact Riemann surface X of genus at–least two. Fix a holomorphic line bundle L over X . Let be the moduli space of Hitchin pairs ( E , φ ∈ H 0 (End 0 ( E ) ⊗ L )) over X of rank r and fixed determinant of degree d . The following conditions are imposed: (i) deg( L ) ≥ 2 g −2, r ≥ 2, and L ⊗ r K X ⊗ r ; (ii) ( r, d ) = 1; and (iii) if g = 2 then r ≥ 6, and if g = 3 then r ≥ 4. We prove that that the isomorphism class of the variety uniquely determines the isomorphism class of the Riemann surface X . Moreover, our analysis shows that is irreducible (this result holds without the additional hypothesis on the rank for low genus).

Hodge polynomials of the SL(2, ℂ)-character variety of an elliptic curve with two marked points

We compute the Hodge polynomials for the moduli space of representations of an elliptic curve with two marked points into SL(2, ℂ). When we fix the conjugacy classes of the representations around the marked points to be diagonal and of modulus one, the character variety is diffeomorphic to the moduli space of strongly parabolic Higgs bundles, whose Betti numbers are known. In that case we can recover some of the Hodge numbers of the character variety. We extend this result to the moduli space of doubly periodic instantons.

Brauer group of moduli spaces of pairs.

We show that the Brauer group of the moduli space of stable pairs with fixed determinant over a curve is zero.

Bohr–Sommerfeld Lagrangians of moduli spaces of Higgs bundles

Abstract Let X be a compact connected Riemann surface of genus at least two. Let M H ( r , d ) denote the moduli space of semistable Higgs bundles on X of rank r and degree d . We prove that the compact complex Bohr–Sommerfeld Lagrangians of M H ( r , d ) are precisely the irreducible components of the nilpotent cone in M H ( r , d ) . This generalizes to Higgs G -bundles and also to the parabolic Higgs bundles.

The topology of parabolic character varieties of free groups

Let

CONNECTION ON PARABOLIC VECTOR BUNDLES OVER CURVES

G

Moduli spaces of parabolic U(p,q)-Higgs bundles

be a complex affine algebraic reductive group, and let