COMPLETE HYPERKÄHLER 4n-MANIFOLDS WITH n COMMUTING TRI-HAMILTONIAN VECTOR FIELDS

We classify those manifolds mentioned in the title which have finite topological type. Namely we show any such connected M is isomorphic to a hyperkähler quotient of a flat quaternionic vector space H by an abelian group. We also show that a compact connected and simply connected 3-Sasakian manifold of dimension 4n− 1 whose isometry group has rank n+1 is isometric to a 3-Sasakian quotient of a sphere by a torus. As a corollary, a compact connected quaternion-Kähler 4n-manifold with positive scalar curvature and isometry group of rank n+ 1 is isometric to HP or Gr2(C). Hyperkähler metrics in dimension 4n with n commuting tri-holomorphic Killing vector fields form a large class of explicitly known Ricci-flat metrics. The flat H, the Taub-NUT, and the Eguchi-Hanson metric are of this form. So are the multiEguchi-Hanson metrics (a.k.a. ALE-spaces of type Ak) and their Taub-NUT-like deformations, both due to Gibbons and Hawking. Higher dimensional examples include the Calabi metrics on T ∗CPn and the asymptotic metrics on moduli spaces of magnetic monopoles. A powerful method of constructing such metrics was given by Lindström and Roček [17] (see also [14, 18]). It associates a hyperkähler metric having n commuting tri-Hamiltonian (meaning Hamiltonian for all three Kähler forms) Killing vector fields to every real-valued function on an open subset of R ⊗Rn which is polyharmonic, i.e. harmonic on any affine 3-dimensional subspace of the form a+R3⊗Rv, a ∈ R ⊗ R, v ∈ R. This construction gives locally all hyperkähler metric with n commuting tri-Hamiltonian (hence Killing) vector fields. On the other hand, the global properties of the resulting metrics in particular completeness have not been investigated. There is another construction that of hyperkähler quotient which, while in this case more restrictive, has the advantage that the global properties of the resulting manifold are accessible. In particular, a hyperkähler quotient of a complete manifold by a compact Lie group is complete. A basic example is the hyperkähler quotient of H × (S × R) by the diagonal circle action which yields the TaubNUT metric. More generally, we obtain complete hyperkähler 4n-manifolds with a tri-Hamiltonian (hence isometric) T -action by taking hyperkähler quotients of quaternionic vector spaces by tori. This class of manifolds has been investigated in detail in [6]. They are closely related to projective toric varieties. In particular, they are diffeomorphic to a union of cotangent bundles of finitely many such varieties glued together. There is an interesting relation to symplectic geometry and Delzant’s theorem [10] which states that a compact symplectic 2n-manifold with a Hamiltonian T action is completely determined, as a Hamiltonian T -manifold, by the image of the moment map. Hyperkähler 4n-manifolds with a tri-Hamiltonian T -action can