C-projective compactification; (quasi-)Kähler metrics and CR boundaries

Abstract

For complete complex connections on almost complex manifolds we introduce a natural definition of compactification. This is based on almost c-projective geometry, which is the almost complex analogue of projective differential geometry. The boundary at infinity is a (possibly non-integrable) CR structure. The theory applies to almost Hermitian manifolds which admit a complex metric connection of minimal torsion, which means that they are quasi-K\"ahler in the sense of Gray-Hervella; in particular it applies to K\"ahler and nearly K\"ahler manifolds. Via this canonical connection, we obtain a notion of c-projective compactification for quasi-K\"ahler metrics of any signature. We describe an asymptotic form for metrics that is necessary and sufficient for c-projective compactness. This metric form provides local examples and, in particular, shows that the usual complete K\"ahler metrics associated to smoothly bounded, strictly pseudoconvex domains in ${\Bbb C}^n$ are c-projectively compact. For a smooth manifold with boundary and a complete quasi-K\"ahler metric $g$ on the interior, we show that if its almost c-projective structure extends smoothly to the boundary then so does its scalar curvature. We prove that in this case (and under some natural assumptions on the extension of $J$ to the boundary) $g$ is almost c-projectively compact if and only if this scalar curvature is non-zero on an open dense set of the boundary. In that case it is, along the boundary, locally constant and hence nowhere zero there. Finally we describe the asymptotics of the curvature, showing, in particular, that the canonical connection satisfies an asymptotic Einstein condition. Key to much of the development is a certain real tractor calculus for almost c-projective geometry, and this is developed in the article.