Moduli of parabolic Higgs bundles and Atiyah algebroids

Abstract

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.

The birational geometry of unramified irregular Higgs bundles on curves

“…The upshot is that it is possible to encode the compatible parabolic structure of the Higgs bundle in terms of a parabolic structure of the spectral sheaf. This is related to the treatment of [17] Proposition 2.2 except that we take a slightly different point of view as them: instead of partitioning the eigenvalues of the Higgs field into the graded pieces of a given filtration, we take the direct sum of the eigenspaces (the fibers of the spectral sheaf over the points lying above a parabolic point of C) and exhaust this space by an increasing filtration.…”

“…However, for explicit computation of symplectic leaves of these holomorphic Poisson spaces, namely moduli spaces of stable parabolic Higgs bundles with fixed semi-simple singular parts, it would be useful to have a refined correspondence that takes into account the fixing of the singular parts too. Such a refined version in the case of regular (or logarithmic) singularities has been provided by M. Logares and J. Martens [17]. However, to our knowledge such a refined BNR-construction in the irregular singular case has not yet been worked out; the purpose of this paper is to give such a refinement.…”

Bohr–Sommerfeld Lagrangians of moduli spaces of Higgs bundles

“…(12) From Lemma 2.1 we have ω(z)(v) = dω(ξ , v). (13) The subvariety L ⊂ M H (r, d) is closed under the action of C * on M H (r, d). Indeed, the Hitchin map H in (10) is C * -equivariant, and the point 0 ∈ V is fixed by the action of C * .…”

“…It is enough to assume that there are stable parabolic Higgs bundles. The reader might check [13] for details on the symplectic structure of these spaces.…”

Introduction to Nonabelian Hodge Theory

“…• the nonabelian Hodge correspondence over quasiprojective varieties, in its tame [52,65,5,43,53,54] and wild [6,7,55,74] variants; • moduli spaces of parabolic Higgs bundles, e.g., [46,58,8,56,23,48];…”