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BY-NC-ND 4.0 license Open Access Published by De Gruyter Open Access February 10, 2017

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

Roger Bielawski
From the journal Complex Manifolds

Abstract

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.

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Received: 2016-12-13
Accepted: 2016-12-13
Published Online: 2017-2-10
Published in Print: 2017-2-23

© 2017

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

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