Arithmetic and metric aspects of open de Rham spaces

TL;DR

This paper computes the motivic class and related polynomials of open de Rham spaces, linking them to quiver varieties and constructing hyperk"ahler metrics, thus advancing understanding of irregular connections and wild character varieties.

Contribution

It provides explicit motivic and polynomial computations for open de Rham spaces and establishes their identification with quiver varieties, also constructing hyperk"ahler metrics.

Findings

Motivic class and weight polynomial computed for open de Rham spaces.

Confirmed the purity conjecture aligning with the mixed Hodge polynomial.

Constructed hyperk"ahler metrics, including expected ALF types in 4D cases.

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.