Hypertoric manifolds and hyperK\"ahler moment maps

TL;DR

This paper classifies complete hyperK"ahler manifolds with torus symmetries, explores properties of their hyperK"ahler moment maps, and introduces new examples, highlighting differences from symplectic cases.

Contribution

It provides a classification of certain hyperK"ahler manifolds, analyzes moment map properties, and constructs new examples of hypertoric manifolds with infinite topological type.

Findings

HyperK"ahler moment maps have connected fibers and are surjective in classified cases.

New hypertoric manifolds of infinite topological type are constructed.

Non-Abelian group actions can produce hyperK"ahler moment maps that are not surjective and have disconnected fibers.

Abstract

We discuss various aspects of moment map geometry in symplectic and hyperK\"ahler geometry. In particular, we classify complete hyperK\"ahler manifolds of dimension $4n$ with a tri-Hamiltonian action of a torus of dimension $n$, without any assumption on the finiteness of the Betti numbers. As a result we find that the hyperK\"ahler moment in these cases has connected fibres, a property that is true for symplectic moment maps, and is surjective. New examples of hypertoric manifolds of infinite topological type are produced. We provide examples of non-Abelian tri-Hamiltonian group actions of connected groups on complete hyperK\"ahler manifolds such that the hyperK\"ahler moment map is not surjective and has some fibres that are not connected. We also discuss relationships to symplectic cuts, hyperK\"ahler modifications and implosion constructions.