Arithmetic and metric aspects of open de Rham spaces

Tamás Hausel; Michael Lennox Wong; Dimitri Wyss

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arXiv · 1807.04057

Arithmetic and metric aspects of open de Rham spaces

Tamás Hausel·Michael Lennox Wong·Dimitri Wyss

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Abstract

In this paper we determine the motivic class--in particular, the weight polynomial and conjecturally the Poincar\'e polynomial--of the open de Rham space, defined and studied by Boalch, of certain moduli spaces of irregular meromorphic connections on the trivial rank $n$ bundle on $\mathbb{P}^1$. 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--Schr\"oer. We finish with constructing natural complete hyperk\"ahler metrics on them, which in the $4$-dimensional cases are expected to be of type ALF.

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Tamás Hausel, Michael Lennox Wong, Dimitri Wyss. 2018-07-11. Arithmetic and metric aspects of open de Rham spaces. https://arxiv.org/abs/1807.04057

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