Compactifications of indefinite 3-Sasaki structures and their quaternionic Kähler quotients
| Authors | |
|---|---|
| Year of publication | 2024 |
| Type | Article in Periodical |
| Magazine / Source | Annali di Matematica Pura ed Applicata |
| MU Faculty or unit | |
| Citation | |
| Web | https://link.springer.com/article/10.1007/s10231-023-01385-0 |
| Doi | http://dx.doi.org/10.1007/s10231-023-01385-0 |
| Keywords | Projective differential geometry; Einstein manifolds; Sasaki manifolds; Hyper-Kähler and quaternionic Kähler geometry; Holonomy; Geometric compactifications |
| Description | We show that 3-Sasaki structures admit a natural description in terms of projective differential geometry. First we establish that a 3-Sasaki structure may be understood as a projective structure whose tractor connection admits a holonomy reduction, satisfying a particular non-vanishing condition, to the (possibly indefinite) unitary quaternionic group Sp(p, q). Moreover, we show that, if a holonomy reduction to Sp(p, q) of the tractor connection of a projective structure does not satisfy this condition, then it decomposes the underlying manifold into a disjoint union of strata including open manifolds with (indefinite) 3-Sasaki structures and a closed separating hypersurface at infinity with respect to the 3-Sasaki metrics. It is shown that the latter hypersurface inherits a Biquard–Fefferman conformal structure, which thus (locally) fibers over a quaternionic contact structure, and which in turn compactifies the natural quaternionic Kähler quotients of the 3-Sasaki structures on the open manifolds. As an application, we describe the projective compactification of (suitably) complete, non-compact (indefinite) 3-Sasaki manifolds and recover Biquard’s notion of asymptotically hyperbolic quaternionic Kähler metrics. |
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