Complete hyperkaehler 4n-manifolds with a local tri-Hamiltonian R^n-action

TL;DR

This paper classifies certain hyperkaehler 4n-manifolds with a specific symmetry, showing they are hyperkaehler quotients of flat quaternionic spaces, and relates 3-Sasakian and quaternion-Kaehler manifolds to known geometric quotients.

Contribution

It provides a classification of hyperkaehler 4n-manifolds with finite topological type as hyperkaehler quotients, and characterizes specific 3-Sasakian and quaternion-Kaehler manifolds as quotients of standard spaces.

Findings

Hyperkaehler 4n-manifolds with finite topological type are hyperkaehler quotients.

Compact 3-Sasakian manifolds with maximal isometry rank are quotients of spheres by tori.

Quaternion-Kaehler manifolds with positive scalar curvature and maximal isometry rank are projective spaces or Grassmannians.

Abstract

We classify those manifolds mentioned in the title which have finite topological type. Namely we show any such connected M is isomorphic to a hyperkaehler quotient of a flat quaternionic vector space 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-Kaehler 4n-manifold with positive scalar curvature and isometry group of rank n+1 is isometric to the quaternionic projective space or the complex grassmanian.