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Arithmetic and metric aspects of open de Rham spaces. (arXiv:1807.04057v1 [math.AG])
来源于:arXiv
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 of irregular
meromorphic connections on the trivial 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. 查看全文>>