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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Autor principal:	Bielawski Roger
Formato:	article
Lenguaje:	EN
Publicado:	De Gruyter 2017
Materias:	Mathematics QA1-939
Acceso en línea:	https://doaj.org/article/7534528826304ae88b8fc1b8cfacd4bd
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spelling	oai:doaj.org-article:7534528826304ae88b8fc1b8cfacd4bd2021-12-02T17:14:47ZSlices to sums of adjoint orbits, the Atiyah-Hitchin manifold, and Hilbert schemes of points2300-744310.1515/coma-2017-0003https://doaj.org/article/7534528826304ae88b8fc1b8cfacd4bd2017-02-01T00:00:00Zhttps://doi.org/10.1515/coma-2017-0003https://doaj.org/toc/2300-7443We 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.Bielawski RogerDe GruyterarticleMathematicsQA1-939ENComplex Manifolds, Vol 4, Iss 1, Pp 16-36 (2017)
institution	DOAJ
collection	DOAJ
language	EN
topic	Mathematics QA1-939
spellingShingle	Mathematics QA1-939 Bielawski Roger Slices to sums of adjoint orbits, the Atiyah-Hitchin manifold, and Hilbert schemes of points
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.
format	article
author	Bielawski Roger
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
publishDate	2017
url	https://doaj.org/article/7534528826304ae88b8fc1b8cfacd4bd
work_keys_str_mv	AT bielawskiroger slicestosumsofadjointorbitstheatiyahhitchinmanifoldandhilbertschemesofpoints
_version_	1718381265507319808

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

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