Slices to sums of adjoint orbits, the Atiyah-Hitchin manifold, and Hilbert schemes of points

Slices to sums of adjoint orbits, the Atiyah-Hitchin manifold, and Hilbert schemes of points

We show that the regular Slodowy slice to the sum of two semisimple adjoint orbits of GL(n, ℂ) is isomorphic to the deformation of the D2-singularity if n = 2, the Dancer deformation of the double cover of the Atiyah-Hitchin manifold if n = 3, and to the Atiyah-Hitchin manifold itself if n = 4. For...

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Main Author: Bielawski Roger
Format: Article
Language:English
Published: De Gruyter 2017-02-01
Series:Complex Manifolds
Online Access:https://doi.org/10.1515/coma-2017-0003
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spelling doaj-7534528826304ae88b8fc1b8cfacd4bd2021-09-06T19:19:41ZengDe GruyterComplex Manifolds2300-74432017-02-0141163610.1515/coma-2017-0003coma-2017-0003Slices to sums of adjoint orbits, the Atiyah-Hitchin manifold, and Hilbert schemes of pointsBielawski Roger0Institut für Differentialgeometrie, Universität Hannover, Welfengarten 1, D-30167, HannoverGermanyWe show that the regular Slodowy slice to the sum of two semisimple adjoint orbits of GL(n, ℂ) is isomorphic to the deformation of the D2-singularity if n = 2, the Dancer deformation of the double cover of the Atiyah-Hitchin manifold if n = 3, and to the Atiyah-Hitchin manifold itself if n = 4. For higher n, such slices to the sum of two orbits, each having only two distinct eigenvalues, are either empty or biholomorphic to open subsets of the Hilbert scheme of points on one of the above surfaces. In particular, these open subsets of Hilbert schemes of points carry complete hyperkähler metrics. In the case of the double cover of the Atiyah-Hitchin manifold this metric turns out to be the natural L2-metric on a hyperkähler submanifold of the monopole moduli space.https://doi.org/10.1515/coma-2017-0003
collection DOAJ
language English
format Article
sources DOAJ
author Bielawski Roger
spellingShingle Bielawski Roger
Slices to sums of adjoint orbits, the Atiyah-Hitchin manifold, and Hilbert schemes of points
Complex Manifolds
author_facet Bielawski Roger
author_sort Bielawski Roger
title Slices to sums of adjoint orbits, the Atiyah-Hitchin manifold, and Hilbert schemes of points
title_short Slices to sums of adjoint orbits, the Atiyah-Hitchin manifold, and Hilbert schemes of points
title_full Slices to sums of adjoint orbits, the Atiyah-Hitchin manifold, and Hilbert schemes of points
title_fullStr Slices to sums of adjoint orbits, the Atiyah-Hitchin manifold, and Hilbert schemes of points
title_full_unstemmed Slices to sums of adjoint orbits, the Atiyah-Hitchin manifold, and Hilbert schemes of points
title_sort slices to sums of adjoint orbits, the atiyah-hitchin manifold, and hilbert schemes of points
publisher De Gruyter
series Complex Manifolds
issn 2300-7443
publishDate 2017-02-01
description We show that the regular Slodowy slice to the sum of two semisimple adjoint orbits of GL(n, ℂ) is isomorphic to the deformation of the D2-singularity if n = 2, the Dancer deformation of the double cover of the Atiyah-Hitchin manifold if n = 3, and to the Atiyah-Hitchin manifold itself if n = 4. For higher n, such slices to the sum of two orbits, each having only two distinct eigenvalues, are either empty or biholomorphic to open subsets of the Hilbert scheme of points on one of the above surfaces. In particular, these open subsets of Hilbert schemes of points carry complete hyperkähler metrics. In the case of the double cover of the Atiyah-Hitchin manifold this metric turns out to be the natural L2-metric on a hyperkähler submanifold of the monopole moduli space.
url https://doi.org/10.1515/coma-2017-0003
work_keys_str_mv AT bielawskiroger slicestosumsofadjointorbitstheatiyahhitchinmanifoldandhilbertschemesofpoints
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