Projective geometry of Sasaki-Einstein structures and their compactification
| Authors | |
|---|---|
| Year of publication | 2019 |
| Type | Article in Periodical |
| Magazine / Source | Dissertationes Mathematicae |
| MU Faculty or unit | |
| Citation | |
| Web | https://www.impan.pl/en/publishing-house/journals-and-series/dissertationes-mathematicae/online/113324/projective-geometry-of-sasaki-einstein-structures-and-their-compactification |
| Doi | http://dx.doi.org/10.4064/dm786-7-2019 |
| Keywords | projective differential geometry; Sasaki-Einstein manifolds; holonomy reductions of Cartan connection; conformal geometry; special contact geometries; Kahler manifolds; CR geometry |
| Description | We show that the standard definitions of Sasaki structures have elegant and simplifying interpretations in terms of projective differential geometry. For Sasaki-Einstein structures we use projective geometry to provide a resolution of such structures into geometrically less rigid components; the latter elemental components are separately, complex, orthogonal, and symplectic holonomy reductions of the canonical projective tractor/Cartan connection. This leads to a characterisation of Sasaki-Einstein structures as projective structures with certain unitary holonomy reductions. As an immediate application, this is used to describe the projective compactification of indefinite (suitably) complete noncompact Sasaki-Einstein structures and to prove that the boundary at infinity is a Fefferman conformal manifold that thus fibres over a nondegenerate CR manifold (of hypersurface type). We prove that this CR manifold coincides with the boundary at infinity for the c-projective compactification of the Kahler-Einstein manifold that arises, in the usual way, as a leaf space for the defining Killing field of the given Sasaki-Einstein manifold. A procedure for constructing examples is given. The discussion of symplectic holonomy reductions of projective structures leads us moreover to a new and simplifying approach to contact projective geometry. This is of independent interest and is treated in some detail. |
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