Twistor quotients of hyperkaehler manifolds

TL;DR

This paper extends the hyperkaehler quotient construction to cases without a group action preserving the entire structure, allowing many known and new hyperkaehler manifolds to be realized as quotients.

Contribution

It introduces a generalized quotient construction for hyperkaehler manifolds using complex group actions on each complex structure, broadening the scope of known hyperkaehler examples.

Findings

All hyperkaehler structures on semisimple coadjoint orbits of complex semisimple Lie groups are realized as quotients.

The generalized Legendre transform construction is incorporated into this framework.

Many new hyperkaehler manifolds are obtained through this generalized quotient approach.

Abstract

We generalize the hyperkaehler quotient construction to the situation where there is no group action preserving the hyperkaehler structure but for each complex structure there is an action of a complex group preserving the corresponding complex symplectic structure. Many (known and new) hyperkaehler manifolds arise as quotients in this setting. For example, all hyperkaehler structures on semisimple coadjoint orbits of a complex semisimple Lie group $G$ arise as such quotients of $T^*G$. The generalized Legendre transform construction of Lindstroem and Rocek is also explained in this framework.