Algebraic families of constant scalar curvature K\"ahler metrics

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.