Nearly K\"ahler Geometry and $(2,3,5)$-Distributions via Projective Holonomy

Abstract

We show that any dimension $6$ nearly K\"ahler (or nearly para-K\"ahler) geometry arises as a projective manifold equipped with a $\textrm{G}_2^{(*)}$ holonomy reduction. In the converse direction we show that if a projective manifold is equipped with a parallel $7$-dimensional cross product on its standard tractor bundle then the manifold is: a Riemannian nearly K\"ahler manifold, if the cross product is definite, otherwise, if the cross product has the other algebraic type, the manifold is in general stratified with nearly K\"ahler and nearly para-K\"ahler parts separated by a hypersurface which canonically carries a Cartan $(2,3,5)$-distribution. This hypersurface is a projective infinity for the pseudo-Riemannian geometry elsewhere on the manifold, and we establish how the Cartan distribution can be understood explicitly, and also in terms of conformal geometry, as a limit of the ambient nearly (para\nobreakdash-)K\"ahler structures. Any real-analytic $(2,3,5)$-distribution is seen to arise as such a limit, because we can solve the geometric Dirichlet problem of building a collar structure equipped with the required holonomy-reduced projective structure. A model geometry for these structures is provided by the projectivization of the imaginary (split) octonions. Our approach is to use Cartan/tractor theory to provide a curved version of this geometry, this encodes a curved version of the algebra of imaginary (split) octonions as a flat structure over its projectivization. The perspective is used to establish detailed results concerning the projective compactification of nearly (para\nobreakdash-)K\"ahler manifolds, including how the almost (para\nobreakdash-)complex structure and metric smoothly degenerate along the singular hypersurface to give the distribution there.