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Publication# Arithmetic and metric aspects of open de Rham spaces

Abstract

In this paper we determine the motivic class---in particular, the weight polynomial and conjecturally the Poincaré polynomial---of the open de Rham space, defined and studied by Boalch, of certain moduli of irregular meromorphic connections on the trivial bundle on $\blackbb{P}^1$. 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--Schröer. We finish with constructing natural complete hyperkähler metrics on them, which in the $4$-dimensional cases are expected to be of type ALF.

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Related publications (1)

Related concepts (11)

Metric space

In mathematics, a metric space is a set together with a notion of distance between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general setting for studying many of the concepts of mathematical analysis and geometry. The most familiar example of a metric space is 3-dimensional Euclidean space with its usual notion of distance. Other well-known examples are a sphere equipped with the angular distance and the hyperbolic plane.

Quiver (mathematics)

In graph theory, a quiver is a directed graph where loops and multiple arrows between two vertices are allowed, in other words a multidigraph. They are commonly used in representation theory: a representation V of a quiver assigns a vector space V(x) to each vertex x of the quiver and a linear map V(a) to each arrow a. In , a quiver can be understood to be the underlying structure of a , but without composition or a designation of identity morphisms. That is, there is a forgetful functor from Cat to Quiv.

Fiber bundle

In mathematics, and particularly topology, a fiber bundle (or, in Commonwealth English: fibre bundle) is a space that is a product space, but may have a different topological structure. Specifically, the similarity between a space and a product space is defined using a continuous surjective map, that in small regions of behaves just like a projection from corresponding regions of to The map called the projection or submersion of the bundle, is regarded as part of the structure of the bundle.

Dimitri Stelio Wyss, Tamás Hausel, Michael Lennox Wong

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 m

2018