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

Roger Bielawski. "Slices to sums of adjoint orbits, the Atiyah-Hitchin manifold, and Hilbert schemes of points." Complex Manifolds 4.1 (2017): 16-36. <http://eudml.org/doc/288009>.

@article{RogerBielawski2017,
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.},
author = {Roger Bielawski},
journal = {Complex Manifolds},
keywords = {Slowdy slice; Atiyah-Hitchin manifold},
language = {eng},
number = {1},
pages = {16-36},
title = {Slices to sums of adjoint orbits, the Atiyah-Hitchin manifold, and Hilbert schemes of points},
url = {http://eudml.org/doc/288009},
volume = {4},
year = {2017},
}

TY - JOUR
AU - Roger Bielawski
TI - Slices to sums of adjoint orbits, the Atiyah-Hitchin manifold, and Hilbert schemes of points
JO - Complex Manifolds
PY - 2017
VL - 4
IS - 1
SP - 16
EP - 36
AB - 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.
LA - eng
KW - Slowdy slice; Atiyah-Hitchin manifold
UR - http://eudml.org/doc/288009
ER -