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Projective geometry of Sasaki–Einstein structures and their compactification

Volume 546 / 2019

A. Rod Gover, Katharina Neusser, Travis Willse Dissertationes Mathematicae 546 (2019), 1-64 MSC: Primary 53A20, 53B10, 53C25, 53C29; Secondary 35Q76, 53A30, 53A40, 53C55. DOI: 10.4064/dm786-7-2019 Published online: 28 November 2019


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 Kähler–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.


  • A. Rod GoverDepartment of Mathematics
    The University of Auckland
    Private Bag 92019
    Auckland 1142, New Zealand
  • Katharina NeusserDepartment of Mathematics and Statistics
    Masaryk University
    Kotlářská 267/2
    611 37 Brno, Czech Republic
  • Travis WillseCentro de Investigación en Matemáticas
    A.P. 402
    Guanajuato, Gto. 36000, Mexico

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