  Asymptotically anti-de Sitter space-times in pure gravity with negative
cosmological constant are described, in all space-time dimensions greater than
two, by classical degrees of freedom on the conformal boundary at space-like
infinity. Their effective boundary action has a conformal anomaly for even
dimensions and is conformally invariant for odd ones. These degrees of freedom
are encoded in traceless tensor fields in the Fefferman-Graham asymptotic
metric for any choice of conformally flat boundary and generate all
Schwarzschild and Kerr black holes in anti-de Sitter space-time. We argue that
these fields describe components of an energy-momentum tensor of a boundary
theory and show explicitly how this is realized in 2+1 dimensions. There, the
Fefferman-Graham fields reduce to the generators of the Virasoro algebra and
give the mass and the angular momentum of the BTZ black holes. Their local
expression is the Liouville field in a general curved background.
