Arithmetic and metric aspects of open de Rham spaces

In this paper, we determine the motivic class - in particular, the weight polynomial and conjecturally the Poincare polynomial - of the open de Rham space, defined and studied by Boalch, of certain moduli spaces of irregular meromorphic connections on the trivial rank n$n$ bundle on P1$\mathbb {P}<^>1$. The computation is by motivic Fourier transform. We show that the result satisfies the purity conjecture, that is, it agrees with the pure part of the conjectured mixed Hodge polynomial of the corresponding wild character variety. We also identify the open de Rham spaces with quiver varieties with multiplicities of Yamakawa and Geiss-Leclerc-Schroer. We finish with constructing natural complete hyperkahler metrics on them, which in the four-dimensional cases are expected to be of type ALF.

Automated summary

In this paper, the motivic class of the open de Rham space on certain moduli spaces is determined, using motivic Fourier transform. The result agrees with the purity conjecture and also identifies the open de Rham spaces with quiver varieties. Additionally, natural complete hyperkähler metrics are constructed on them, expected to be of type ALF in four-dimensional cases.