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# Surveys in Differential Geometry

## Volume 19 (2014)

### Algebraic families of constant scalar curvature Kähler metrics

Pages: 111 – 137

DOI: http://dx.doi.org/10.4310/SDG.2014.v19.n1.a5

#### Author

#### Abstract

We give a new proof of the fact that the condition of a Fano manifold admitting a Kähler-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 Kähler metrics.

#### 2010 Mathematics Subject Classification

53C55