On some Moment Maps and Induced Hopf Bundles in the Quaternionic Projective Space

TL;DR

This paper explores the geometry of quaternionic projective spaces through moment maps, relating zero sets to focal and Sasakian-Einstein structures, and constructs new complex structures on Stiefel manifolds.

Contribution

It introduces a diagram linking moment map zero sets to geometric structures and constructs novel complex structures on specific Stiefel manifolds.

Findings

Zero sets of moment maps relate to focal and Sasakian-Einstein structures.

Constructed a complex structure on V_2(C^{n+1}) and V_4(R^{n+1}).

New complex structure on V_2 is not compatible with hypercomplex structure.

Abstract

We describe a diagram containing the zero sets of the moment maps associated to the diagonal U(1) and Sp(1) actions on the quaternionic projective space HP^n. These sets are related both to focal sets of submanifolds and to Sasakian-Einstein structures on induced Hopf bundles. As an application, we construct a complex structure on the Stiefel manifolds V_2 (C^{n+1}) and V_4 (R^{n+1}), the one on the former manifold not being compatible with its known hypercomplex structure.