Skip to content
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


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.


[1] M.F. Atiyah and N.J. Hitchin, The geometry and dynamics of magnetic monopoles, Princeton University Press, Princeton (1988).10.1515/9781400859306Search in Google Scholar

[2] A. Beauville, ‘Variétés Kähleriennes dont la première classe de Chern est nulle’, J. Differential Geom. 18 (1983), no. 4, 755-782.Search in Google Scholar

[3] A.L. Besse, Einstein manifolds, Springer, Berlin, 1987.10.1007/978-3-540-74311-8Search in Google Scholar

[4] R. Bielawski, ‘Hyperkähler structures and group actions’, J. London Math. Soc. 55 (1997), 400-414.10.1112/S0024610796004723Search in Google Scholar

[5] R. Bielawski, ‘Invariant hyperkähler metrics with a homogeneous complex structure’, Math. Proc. Cambridge Philos. Soc. 122 (1997), 473-482.10.1017/S0305004197001953Search in Google Scholar

[6] R. Bielawski, ‘On the hyperkähler metrics associated to singularities of nilpotent varieties’, Ann. Glob. Anal. Geom. 14 (1996), 177-191.10.1007/BF00127972Search in Google Scholar

[7] R. Bielawski, ‘Monopoles and the Gibbons-Manton metric’, Comm. Math. Phys. 194 (1998), 297-321.10.1007/s002200050359Search in Google Scholar

[8] R. Bielawski, ‘Asymptotic Metrics for SU(N)-Monopoles with Maximal Symmetry Breaking’, Comm. Math. Phys. 199 (1998), 297-325.10.1007/s002200050503Search in Google Scholar

[9] O. Biquard, ‘Sur les équations de Nahm et les orbites coadjointes des groupes de Lie semi-simples complexes’, Math. Ann. 304 (1996), 253-276.10.1007/BF01446293Search in Google Scholar

[10] G. Chalmers, ‘Implicit metric on a deformation of the Atiyah-Hitchin manifold’, Phys. Rev. D 58 (1998), no. 12, 125011, 7 pp.10.1103/PhysRevD.58.125011Search in Google Scholar

[11] S.A. Cherkis and A. Kapustin, ‘Dk gravitational instantons and Nahm equations’, Adv. Theor. Math. Phys. 2 (1998), no. 6, 1287-1306 (1999).Search in Google Scholar

[12] A. S. Dancer, ‘Nahm’s equations and hyper-Kähler geometry’, Comm. Math. Phys. 158 (1993), 545-568.10.1007/BF02096803Search in Google Scholar

[13] S.K. Donaldson and P.B. Kronheimer, The geometry of four-manifolds, Clarendon Press, Oxford, (1990).Search in Google Scholar

[14] R. Hartshorne, ‘The genus of space curves’, Ann. Univ. Ferrara. Sez. VII, 40 (1994), 207-223.10.1007/BF02834521Search in Google Scholar

[15] T. Hausel, M. Wong and D. Wyss, ‘Arithmetic and metric aspects of open de Rham spaces’, in preparation.Search in Google Scholar

[16] N.J. Hitchin, ‘Twistor construction of Einstein metrics’, in: Global Riemannian geometry (Durham, 1983), 115-125, Horwood, Chichester, 1984.Search in Google Scholar

[17] N.J. Hitchin, N.S. Manton, and M.K. Murray, ‘Symmetric monopoles’, Nonlinearity 8 (1995), no. 5, 661-692.Search in Google Scholar

[18] J.C. Hurtubise, ‘The classification of monopoles for the classical groups’, Comm. Math. Phys. 120 (1989), 613-641.10.1007/BF01260389Search in Google Scholar

[19] C. Jackson, ‘Nilpotent slices and Hilbert schemes’, Ph-D. Thesis, University of Chicago (2007), 38 pp.Search in Google Scholar

[20] P.Z. Kobak and A. Swann, ‘Classical nilpotent orbits as hyper-K¨hler quotients’, Internat. J. Math. 7 (1996), 193-210.10.1142/S0129167X96000116Search in Google Scholar

[21] A.G. Kovalev, ‘Nahm’s equations and complex adjoint orbits’, Quart. J. Math. Oxford Ser. (2) 47 (1996), 41-58.10.1093/qmath/47.1.41Search in Google Scholar

[22] P.B. Kronheimer, ‘A hyper-kählerian structure on coadjoint orbits of a semisimple complex group’, J. London Math. Soc. 42 (1990), 193-208.10.1112/jlms/s2-42.2.193Search in Google Scholar

[23] P.B. Kronheimer, ‘Instantons and the geometry of the nilpotent variety’, J. Differential Geom. 32 (1990), 473-490.10.4310/jdg/1214445316Search in Google Scholar

[24] C. Manolescu, ‘Nilpotent slices, Hilbert schemes, and the Jones polynomial’, Duke Math. J. 132 (2006), 311-369.10.1215/S0012-7094-06-13224-6Search in Google Scholar

[25] H. Nakajima, ‘Instantons on ALE spaces, quiver varieties, and Kac-Moody algebras’, Duke Math. J. 76 (1994), 365-416.10.1215/S0012-7094-94-07613-8Search in Google Scholar

[26] H. Nakajima, Lectures on Hilbert schemes of points on surfaces, American Mathematical Society, Providence, RI, 1999.10.1090/ulect/018Search in Google Scholar

[27] P. Seidel and I. Smith, ‘A link invariant from the symplectic geometry of nilpotent slices’, Duke Math. J. 134 (2006), 45-514.10.1215/S0012-7094-06-13432-4Search in Google Scholar

[28] P. Slodowy, Conjugacy classes in algebraic groups, Lecture Notes in Mathematics 815, Springer, Berlin, 1980.Search in Google Scholar

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.

Downloaded on 1.2.2023 from
Scroll Up Arrow