Description
I will discuss symplectic leaves in moduli spaces of meromorphic $GL_r$-Higgs bundles on a smooth projective curve. These moduli spaces carry natural Poisson structures, studied independently by Bottacin and Markman, and their symplectic leaves are expected to be governed by the adjoint orbits of the residues at the marked points. However, it has not been clear whether the corresponding loci of Higgs bundles are connected. The Hitchin map is also expected to restrict to a symplectic integrable system on each leaf. For general choices of residue orbits, especially non-maximal ones, this leads to the problem of describing the Hitchin base and the generic fibers. I will discuss these questions using $\xi$-parabolic Higgs bundles, which lift the condition of fixing residue orbits to compatible parabolic flag data. This is joint work with Jiachoon Lee.