An attempt is made of giving a self-contained introduction to holomorphic ideas in general relativity, following work over the last thirty years by several authors. The main topics are complex manifolds, two-component spinor calculus, conformal gravity, α-planes in Minkowski space-time, α-surfaces and twistor geometry, anti-self-dual space-times and Penrose transform, spin-3/2 potentials, heaven spaces and heavenly equations.
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