Projective geometry of Sasaki-Einstein structures and their compactification

TL;DR

This paper explores the projective geometric interpretation of Sasaki-Einstein structures, providing new characterizations, compactification methods, and insights into their boundary structures, with applications to CR and Kähler-Einstein geometries.

Contribution

It introduces a projective geometric framework for Sasaki-Einstein structures, characterizes them via holonomy reductions, and describes their compactifications and boundary structures.

Findings

Sasaki-Einstein structures correspond to projective structures with specific unitary holonomy.

The boundary at infinity of certain Sasaki-Einstein structures is a Fefferman conformal manifold.

A new approach to contact projective geometry simplifies the understanding of symplectic holonomy reductions.

Abstract

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 non-compact 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 K\"ahler-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.