In a landmark paper  of the 1970’s, Roger Penrose described a remarkable and unexpected connection between Einstein’s equations and the theory of complex manifolds. The twistor correspondence which he detailed there gives a way of producing all self-dual complex solutions of these equations in terms of the global structure of an associated complex manifold called the twistor space. Unfortunately, the most interesting solutions from the view-point of physics are not the self-dual ones, but rather those of Lorentz signature. If this had been the end of the story, one might have therefore been tempted to conclude that this correspondence was merely a mathematical curiosity with no bearing on physics. Fortunately, however, the Penrose twistor correspondence is but one aspect of a rather more complicated story, which I will endeavor to recount here. Indeed, it turns out that the general complex solution of the 4-dimensional Einstein equations can also be described in terms of complex deformation theory, albeit in terms of non-reduced complex spaces rather than complex manifolds.
Line Bundle Complex Manifold Twistor Space Null Geodesic Conformal Class
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