  We give a new proof of the fact that the condition of a Fano manifold
admitting a K\"ahler-Einstein metric is Zariski-open (provided that the
automorphism group is discrete). This proof does not use the characterisation
involving stability. The arguments involve estimates of Futaki invariants
obtained from a differential-geometric "volume estimate", and variants of the
algebro-geometric arguments of Stoppa. Many of the ideas apply to constant
scalar curvature K\"ahler metrics.
