Publication details

Projective geometry of Sasaki-Einstein structures and their compactification

Authors

GOVER A. Rod NEUSSER Katharina WILLSE Travis

Year of publication 2019
Type Article in Periodical
Magazine / Source Dissertationes Mathematicae
MU Faculty or unit

Faculty of Science

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.
Related projects:

You are running an old browser version. We recommend updating your browser to its latest version.

More info