Roger Bielawski - Tn-invariant hyperkähler 4n-manifolds

Next: Charles P. Boyer - Up: Relativity and Geometry / Previous: Relativity and Geometry /

Roger Bielawski - Tⁿ-invariant hyperkähler 4n-manifolds

	ROGER BIELAWSKI, Max Planck Institute, Bonn, Germany
	Tⁿ-invariant hyperkähler 4n-manifolds

We prove that a connected simply connected complete hyperkähler 4n-manifold of finite topological type equipped with an effective tri-Hamiltonian action of Tⁿ is isometric to a hyperkähler quotient of a flat $\bf H^{d+m}$ by $T^{d-n}\times \bf R^m$ . As a consequence, such manifolds are classified, up to canonical deformations by arrangements of codimension 3 affine subspaces in $\bf R^{3n}$ .

We show a similar result for two other classes of Einstein manifolds: compact 3-Sasakian and compact quaternion-Kähler with positive scalar curvature. In the latter case, we are able to conclude that such a 4n-manifold admitting a group of isometries of rank n+1 is either $\bf H P^n$ or $\textrm{Gr}_2(\bf C^{n+2})$ .

eo@camel.math.ca