The ambient obstruction tensor and the conformal deformation complex

Abstract

The (Fefferman-Graham) ambient obstruction tensor is a conformally invariant symmetric trace-free 2-tensor on even-dimensional Riemannian and pseudo-Riemannian manifolds. The conformal deformation complex is a differential complex related to infinitesimal deformations of conformal structure. We construct a conformally invariant differential operator on algebraic Weyl tensors that gives special curved analogues of certain operators related to the deformation complex and that, upon application to the Weyl curvature, yields the obstruction tensor. This new definition of the obstruction tensor leads to simple direct proofs that the obstruction tensor is divergence free and vanishes for a class of metrics which (strictly) includes the class of conformally Einstein metrics. Our main constructions are based on the ambient metric of Fefferman-Graham and its relation to the conformal tractor connection. We prove that the obstruction tensor is an obstruction to the formal problem of obtaining a Ricci-flat ambient metric and that it may equivalently be viewed as the obstruction to finding an ambient metric with curvature harmonic for a certain (ambient) form Laplacian. This leads to a new ambient formula for the obstruction in terms of a power of this form Laplacian acting on the ambient curvature. This result leads us to construct Laplacian type operators that generalise the conformal Laplacians of Graham-Jenne-Mason-Sparling. As background to these issues, we give an explicit construction of the deformation complex in dimensions n>3, construct two related (detour) complexes, and establish essential properties of the operators in these.