Untwisted and twisted open de Rham spaces

 An "open de Rham space" refers to a moduli space of meromorphic connections on the projective line with underlying trivial bundle.  In the case where the connections have simple poles, it is well-known that these spaces exhibit hyperkähler metrics and can be realized as quiver varieties.  This story can in fact be extended to the case of higher order poles, at least in the "untwisted" case.  The "twisted" spaces, introduced by Bremer and Sage, refer to those which have normal forms diagonalizable only after passing to a ramified cover.  These spaces often arise as quotients by unipotent groups and in some low-dimensional examples one finds some well-known hyperkähler manifolds, such as the moduli of magnetic monopoles.  This is a report on ongoing work with Tamás Hausel and Dimitri Wyss.