Symplectic leaves of meromorphic Hitchin systems

The moduli space of meromorphic Higgs bundles admits a Poisson structure due to the independent work of Bottacin and Markman.

In this paper, we revisit the symplectic leaves of this Poisson structure for the tame case.

We study the partial compactification of the restricted Hitchin map on the symplectic leaves to an algebraically completely integrable system.

In particular, we show that such a partial compactification is realized by the moduli spaces of $\vec{\xi}$-parabolic Higgs bundles.

These same moduli spaces also provide a symplectic resolution of the normalization of the closure of the corresponding symplectic leaves.

Finally, we discuss connectedness results for the corresponding Betti moduli spaces under the tame non-abelian Hodge correspondence.